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The Annals of Statistics
0, Vol. 0, No. 00, 1–38
DOI: 10.1214/11-AOS901
© Institute of Mathematical Statistics, 0
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY
PROCESSES VIA COMPOSITE CHARACTERISTIC FUNCTION
ESTIMATION
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B Y D ENIS B ELOMESTNY
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Duisburg-Essen University
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In this article, the problem of semi-parametric inference on the parameters of a multidimensional Lévy process Lt with independent components
based on the low-frequency observations of the corresponding time-changed
Lévy process LT (t) , where T is a nonnegative, nondecreasing real-valued
process independent of Lt , is studied. We show that this problem is closely
related to the problem of composite function estimation that has recently got
much attention in statistical literature. Under suitable identifiability conditions, we propose a consistent estimate for the Lévy density of Lt and derive
the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax
sense over suitable classes of time-changed Lévy models. Finally, we present
a simulation study showing the performance of our estimation algorithm in
the case of time-changed Normal Inverse Gaussian (NIG) Lévy processes.
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1. Introduction. The problem of nonparametric statistical inference for jump
processes or more generally for semimartingale models has long history and goes
back to the works of Rubin and Tucker (1959) and Basawa and Brockwell (1982).
In the past decade, one has witnessed the revival of interest in this topic which
is mainly related to a wide availability of financial and economical time series
data and new types of statistical issues that have not been addressed before. There
are two major strands of recent literature dealing with statistical inference for
semimartingale models. The first type of literature considers the so-called highfrequency setup, where the asymptotic properties of the corresponding estimates
are studied under the assumption that the frequency of observations tends to infinity. In the second strand of literature, the frequency of observations is assumed to
be fixed (the so-called low-frequency setup) and the asymptotic analysis is done
under the premiss that the observational horizon tends to infinity. It is clear that
none of the above asymptotic hypothesis can be perfectly realized on real data and
they can only serve as a convenient approximation, as in practice the frequency of
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Received December 2010; revised April 2011.
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1 Supported in part by SFB 649 “Economic Risk.”
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MSC2010 subject classifications. Primary 62F10; secondary 62J12, 62F25, 62H12.
Key words and phrases. Time-changed Lévy processes, dependence, pointwise and uniform rates
of convergence, composite function estimation.
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St = S0 exp LT (t) ,
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where L is a Lévy process and T (t) is a time change of the form
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D. BELOMESTNY
observations and the horizon are always finite. The present paper studies the problem of statistical inference for a class of semimartingale models in low-frequency
setup.
Let X = (Xt )t≥0 be a stochastic process valued in Rd and let T = (T (s))s≥0 be
a nonnegative, nondecreasing stochastic process not necessarily independent of X
with T (0) = 0. A time-changed process Y = (Ys )s≥0 is then defined as Ys = XT (s) .
The process T is usually referred to as time change. Even in the case of the
one-dimensional Brownian motion X, the class of time-changed processes XT is
very large and basically coincides with the class of all semimartingales [see, e.g.,
Monroe (1978)]. In fact, the construction in Monroe (1978) is not direct, meaning
that the problem of specification of different models with the specific properties
remains an important issue. For example, the base process X can be assumed to
possess some independence property (e.g., X may have independent components),
whereas a nonlinear time change can induce deviations from the independence.
Along this line, the time change can be used to model dependence for stochastic
processes. In this work, we restrict our attention to the case of time-changed Lévy
processes, that is, the case where X = L is a multivariate Lévy process and T is an
independent of L time change. Time-changed Lévy processes are one step further
in increasing the complexity of models in order to incorporate the so-called stylized features of the financial time series, like volatility clustering [for more details,
see Carr et al. (2003)]. This type of processes in the case of the one-dimensional
Brownian motion was first studied by Bochner (1949). Clark (1973) introduced
Bochner’s time-changed Brownian motion into financial economics: he used it to
relate future price returns of cotton to the variations in volume during different
trading periods. Recently, a number of parametric time-changed Lévy processes
have been introduced by Carr et al. (2003), who model the stock price St by a
geometric time-changed Lévy model
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(1.1)
T (t) =
t
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ρ(u) du
0
with {ρ(u)}u≥0 being a positive mean-reverting process. Carr et al. (2003) proposed to model ρ(u) via the Cox–Ingersoll–Ross (CIR) process. Taking different
parametric Lévy models for L (such as the normal inverse Gaussian or the variance
Gamma processes) results in a wide range of processes with rather rich volatility
structure (depending on the rate process ρ) and various distributional properties
(depending on the specification of L). From statistical point of view, any parametric model (especially one using only few parameters) is prone to misspecification
problems. One approach to deal with the misspecification issue is to adopt the
general nonparametric models for the functional parameters of the underlying process. This may reduce the estimation bias resulting from an inadequate parametric
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
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model. In the case of time-changed Lévy models, there are two natural nonparametric parameters: Lévy density ν, which determines the jump dynamics of the
process L and the marginal distribution of the process T .
In this paper, we study the problem of statistical inference on the characteristics of a multivariate Lévy process L with independent components based on
low-frequency observations of the time-changed process Yt = LT (t) , where T (t)
is a time change process independent of L with strictly stationary increments. We
assume that the distribution of T (t) is unknown, except of its mean value. This
problem is rather challenging one and has not been yet given attention in the literature, except of the special case of T (t) ≡ t [see, e.g., Neumann and Reiß (2009)
and Comte and Genon-Catalot (2010)]. In particular, the main difficulty in constructing nonparametric estimates for the Lévy density ν of L lies in the fact that
the jumps are unobservable variables, since in practice only discrete observations
of the process Y are available. The more high-frequent are the observations, the
more relevant information about the jumps of the underlying process and hence,
about the Lévy density ν are contained in the sample. Such high-frequency based
statistical approach has played a central role in the recent literature on nonparametric estimation for Lévy type processes. For instance, under discrete observations
of a pure Lévy process Lt at times tj = j , j = 0, . . . , n, Woerner (2003) and
Figueroa-López (2004) proposed the quantity
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1 )=
β(f
f (Ltk − Ltk−1 )
n k=1
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β(f ) =
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f (x)ν(x) dx,
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where f is a given “test function.” Turning back to the time-changed Lévy processes, it was shown in Figueroa-López (2009) [see also Rosenbaum and Tankov
(2010)] that in the case, where the rate process ρ in (1.1) is a positive ergodic dif ) is still a consistent estimator for
fusion independent of the Lévy process L, β(f
β(f ) up to a constant, provided the time horizon n and the sampling frequency
−1 converge to infinite at suitable rates. In the case of low-frequency data ( is
fixed), we cannot be sure to what extent the increment Ltk − Ltk−1 is due to one or
several jumps or just to the diffusion part of the Lévy process so that at first sight
it may appear surprising that some kind of inference in this situation is possible at
all. The key observation here is that for any bounded “test function” f
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as a consistent estimator for the functional
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(1.2)
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f LT (tj ) − LT (tj −1 ) → Eπ f LT () ,
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n → ∞,
provided the sequence T (tj ) − T (tj −1 ), j = 1, . . . , n, is stationary and ergodic
with the invariant stationary distribution π . The limiting expectation in (1.2) is
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D. BELOMESTNY
then given by
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Eπ f LT () =
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exp(iu z), u
(1.3)
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E exp iu LT () =
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∞
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2
E[f (Ls )]π(ds).
∞
0
Rd ,
ψ(u) = t −1 log[E exp(iuLt )]
is the characteristic exponent of the Lévy prowhere
cess L and L is the Laplace transform of π . In fact, the most difficult part of
estimation procedure comes only now and consists in reconstructing the characteristics of the underlying Lévy process L from an estimate for L (−ψ(u)). As
we will see, the latter statistical problem is closely related to the problem of composite function estimation, which is known to be highly nonlinear and ill-posed.
The identity (1.3) also reveals the major difference between high-frequency and
low-frequency setups. While in the case of high-frequency data one can directly
estimate linear functionals of the Lévy measure ν, under low-frequency observations, one has to deal with nonlinear functionals of ν rendering the underlying estimation problem nonlinear and ill-posed. The last but not the least, the increments
of time-changed Lévy processes are not any longer independent, hence advanced
tools from time series analysis have to be used for the estimation of L (−ψ(u)).
The paper is organized as follows. In Section 2.1, we introduce the main object of our study, the time-changed Lévy processes. In Section 2.2, our statistical
problem is formulated and its connection to the problem of composite function
estimation is established. In Section 2.3, we impose some restrictions on the structure of the time-changed Lévy processes in order to ensure the identifiability and
avoid the “curse of dimensionality.” Section 3 contains the main estimation procedure. In Section 4, asymptotic properties of the estimates defined in Section 3
are studied. In particular, we derive uniform and pointwise rates of convergence
(Sections 4.3 and 4.4, resp.) and prove their optimality over suitable classes of
time-changed Lévy models (Section 4.5). Section 4.7 contains some discussion.
Finally, in Section 5 we present a simulation study. The rest of the paper contains
proofs of the main results and some auxiliary lemmas. In particular, in Section 7.3
a useful inequality on the probability of large deviations for empirical processes in
uniform metric for the case of weakly dependent random variables can be found.
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2. Main setup.
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exp(tψ(u))π(dt) = L (−ψ(u)),
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∈
and using the independence of L
Taking f (z) = fu (z) =
and T , we arrive at the following representation for the c.f. of LT (s) :
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2.1. Time-changed Lévy processes. Let Lt be a d-dimensional Lévy process
on the probability space (, F , P) with the characteristic exponent ψ(u), that is,
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
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We know by the Lévy–Khintchine formula that
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(2.1)
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ψ(u) = iμ u − u u +
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(|y| ∧ 1)ν(dy) < ∞.
T (t) = bt +
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t ∞
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yρ(dy, ds),
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t
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ρ(s− ) ds,
where ρ is the instantaneous activity rate which is assumed to be nonnegative.
When Lt is the Brownian motion and ρ is proportional to the instantaneous variance rate of the Brownian motion, then Yt is a pure jump Lévy process with the
Lévy measure proportional to ρ. Let us now compute the characteristic function
of Yt . Since T (t) and Lt are independent, we get
(2.3)
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where b is a drift and ρ is the counting measure of jumps in the time change. As in
Carr and Wu (2004), one can take bt = 0 and consider locally deterministic time
changes
(2.2)
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A triplet (μ, , ν) is usually called a characteristic triplet of the d-dimensional
Lévy process Lt .
Let t → T (t), t ≥ 0 be an increasing right-continuous process with left limits
such that T (0) = 0 and for each fixed t, the random variable T (t) is a stopping
time with respect to the filtration F . Suppose furthermore that T (t) is finite P-a.s.
for all t ≥ 0 and that T (t) → ∞ as t → ∞. Then the family of (T (t))t≥0 defines
a random time change. Now consider a d-dimensional process Yt := LT (t) . The
process Yt is called the time-changed Lévy process. Let us look at some examples.
If T (t) is a Lévy process, then Yt would be another Lévy process. A more general
situation is when T (t) is modeled by a nondecreasing semimartingale
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Rd \{0}
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iu y
e
− 1 − iu y · 1{|y|≤1} ν(dy),
is a positive-semidefinite symmetric d × d matrix and ν is a
where μ ∈
Lévy measure on Rd \ {0} satisfying
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φY (u|t) = E eiu LT (t) = Lt (−ψ(u)),
where Lt is the Laplace transform of T (t):
Lt (λ) = E e−λT (t) .
2.2. Statistical problem. In this paper, we are going to study the problem
of estimating the characteristics of the Lévy process L from low-frequency observations Y0 , Y , . . . , Yn of the process Y for some fixed > 0. Moving to
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D. BELOMESTNY
the spectral domain and taking into account (2.1), we can reformulate our problem as the problem of semi-parametric estimation of the characteristic exponent ψ under structural assumption (2.1) from an estimate of φY (u|) based
on Y0 , Y , . . . , Yn . The formula (2.3) shows that the function φY (u|) can be
viewed as a composite function and our statistical problem is hence closely related to the problem of statistical inference on the components of a composite
function. The latter type of problems in regression setup has got much attention
recently [see, e.g., Horowitz and Mammen (2007) and Juditsky, Lepski and Tsybakov (2009)]. Our problem has, however, some features not reflected in the previous literature. First, the unknown link function L , being the Laplace transform
of the r.v. T (), is completely monotone. Second, the complex-valued function ψ
is of the form (2.1) implying, for example, a certain asymptotic behavior of ψ(u)
as u → ∞. Finally, we are not in regression setup and φY (u|) is to be estimated
by its empirical counterpart
φ(u)
=
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n
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eiu (Yj −Y(j −1) ) .
n j =1
The contribution of this paper to the literature on composite function estimation is
twofold. On the one hand, we introduce and study a new type of statistical problems which can be called estimation of a composite function under structural constraints. On the other hand, we propose new and constructive estimation approach
which is rather general and can be used to solve other open statistical problems of
this type. For example, one can directly adapt our method to the problem of semiparametric inference in distributional Archimedian copula-based models [see, e.g.,
McNeil and Nešlehová (2009) for recent results], where one faces the problem of
estimating a multidimensional distribution function of the form
F (x1 , . . . , xd ) = G f1 (x1 ) + · · · + fd (xd ) ,
(x1 , . . . , xd ) ∈ Rd ,
with a completely monotone function G and some functions f1 , . . . , fd . Further
discussion on the problem of composite function estimation can be found in Remark 4.14.
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2.3. Specification analysis. It is clear that without further restrictions on the
class of time-changed Lévy processes our problem of estimating ν is not well defined, as even in the case of the perfectly known distribution of the process Y
the parameters of the Lévy process L are generally not identifiable. Moreover,
the corresponding statistical procedure will suffer from the “curse of dimensionality” as the dimension d increases. In order to avoid these undesirable features, we
have to impose some additional restrictions on the structure of the time-changed
process Y . In statistical literature, one can basically find two types of restricted
composite models: additive models and single-index models. While the latter class
of models is too restrictive in our situation, the former one naturally appears if one
assumes the independence of the components of Lt . In this paper, we study a class
of time-changed Lévy processes satisfying the following two assumptions:
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
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(ALI) The Lévy process Lt has independent components such that at least two
of them are nonzero, that is,
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where ψk , k = 1, . . . , d, are the characteristic exponents of the components of Lt
of the form
(2.5)
ψk (u) = iμk u − σk2 u2 /2
+
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k = 1, . . . , d,
|μl | + σl2 +
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n
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−∂ul ul φ(u)|
Yj,l − Y(j −1),l > 0
u=0 =
n j =1
for at least two different indexes l. Let us make a few remarks on the onedimensional case, where
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Discussion. The advantage of the modeling framework (2.4) is twofold. On the
one hand, models of this type are rather flexible: the distribution of Yt for a fixed t
is in general determined by d + 1 nonparametric components and 2 × d parametric
ones. On the other hand, these models remain parsimonious and, as we will see
later, admit statistical inference not suffering from the “curse of dimensionality”
as d becomes large. The latter feature of our model is in accordance with the
well documented behavior of the additive models in regression setting and may
become particularly important if one is going to use it, for instance, to model large
portfolios of assets. The nondegeneracy assumption (2.6) basically excludes onedimensional models and is not restrictive since it can be always checked prior to
estimation by testing that
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x 2 νl (dx) = 0
for at least two different indexes l.
(ATI) The time change process T is independent of the Lévy process L and
satisfies E[T (t)] = t.
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and
(2.6)
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e − 1 − iux · 1{|x|≤1} νk (dx),
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(2.4)
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(2.7)
φY (u|t) = Lt (−ψ1 (u)),
t ≥ 0.
If L is known, that is, the distribution of the r.v. T () is known, we can consistently estimate the Lévy measure ν1 by inverting L (see Section 4.6 for more
details). In the case when the function L is unknown, one needs some additional
assumptions (e.g., absolute continuity of the time change) to ensure identifiability.
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D. BELOMESTNY
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Indeed, consider a class of the one-dimensional Lévy processes of the so-called
compound exponential type with the characteristic exponent of the form
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ψ(u) = log
1 − ψ(u)
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is completely monotone with L
(0) = 1
As can be easily seen, the function L
(0) = L (0). Moreover, it is fulfilled L
(−ψ(u))
and L
= L (−ψ(u)) for all
u ∈ R. The existence of the time change (increasing) process T with a given
marginal T () can be derived from the general theory of stochastic partial ordering [see Kamae and Krengel (1978)]. The above construction indicates that the
assumption E[T (t)] = t, t ≥ 0, is not sufficient to ensure the identifiability in the
case of one-dimensional time-changed Lévy models.
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ψk (u) = −σk2 −
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k = 1, . . . , d.
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R
eiux ν̄k (x) dx.
For the sake of simplicity, in the sequel we will make the following assumption:
(ALS) The diffusion volatilities σk , k = 1, . . . , d, of the Lévy process L are
supposed to be known.
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By differentiating ψk two times, we get
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ν̄k (x) = x νk (x),
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x 2 νk (x) dx < ∞,
Consider the functions
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3.1. Main ideas. Assume that the Lévy measures of the component processes
1
Lt , . . . , Ldt are absolutely continuous with integrable densities ν1 (x), . . . , νd (x)
that satisfy
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3. Estimation.
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(z) = L log(1 + z) .
L
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is the characteristic exponent of another one-dimensional Lévy prowhere ψ(u)
cess Lt . It is well known [see, e.g., Section 3 in Chapter 4 of Steutel and van Harn
(2004)] that exp(ψ(u)) is the characteristic function of some infinitely divisible
does. Introduce
distribution if exp(ψ(u))
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A way how to extend our results to the case of the unknown (σk ) is outlined in
Section 4.6. Introduce the functions ψ̄k (u) = ψk (u) + σk2 u2 /2 to get
(3.1)
F[ν̄k ](u) = −ψ̄k (u) = −σk2 − ψk (u),
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
where F[ν̄k ](u) stands for the Fourier transform of ν̄k . Denote Z = Y , φk (u) =
∂uk φZ (u), φkl (u) = ∂uk ul φZ (u) and φj kl (u) = ∂uj uk ul φZ (u) for j, k, l ∈ {1, . . . , d}
with
4
5
6
7
(3.2)
10
11
13
φk (u(k) )
=
φl (u(k) )
18
19
20
21
22
23
(3.5)
27
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
(3.6)
10
11
13
14
15
φkk (u
ψk (u) = −1 φl (0)
ψk (u) = −1 φll (0)
9
12
16
17
18
in the case μl = 0. The identities φl (0) = −ψl (0)L (0) and φll (0) = [ψl (0)]2 ×
L (0) − ψl (0)L (0) imply ψl (0) = −[L (0)]−1 φl (0) = −1 φl (0) and ψl (0) =
−[L (0)]−1 φll (0) = −1 φll (0) if ψl (0) = 0, since L (0) = −E[T ()] = −.
Combining this with (3.3) and (3.4), we derive
26
28
ψk (u)
,
ψl (0)
ψk (u)
φk (u(k) )
=
φll (u(k) ) ψl (0)
(3.4)
24
25
8
if μl = 0 and
16
17
7
with u being placed at the kth coordinate of the vector u(k) . Choose some l = k,
such that the component Llt is not degenerated. Then we get from (3.2)
14
15
6
u(k) = (0, . . . , 0, u, 0, . . . , 0) ∈ Rd
(3.3)
3
5
Fix some k ∈ {1, . . . , d} and for any real number u introduce a vector
12
2
4
φZ (u) = E[exp(iu Z)] = L −ψ1 (u1 ) − · · · − ψd (ud ) .
8
9
1
(k) )φ (u(k) ) − φ (u(k) )φ (u(k) )
l
k
lk
,
2
(k)
φl (u )
φkk (u(k) )φll (u(k) ) − φk (u(k) )φllk (u(k) )
,
φll2 (u(k) )
19
20
21
22
23
24
μl = 0,
25
26
27
μl = 0.
Note that in the above derivations we have repeatedly used the assumption (ATI),
that turns out to be crucial for the identifiability. The basic idea of the algorithm,
we shall develop in the Section 3.2, is to estimate ν̄k by an application of the
regularized Fourier inversion formula to an estimate of ψ̄k (u). As indicated by
formulas (3.5) and (3.6), one could, for example, estimate ψ̄k (u), if some estimates
for the functions φk (u), φlk (u) and φllk (u) are available.
R EMARK 3.1. One important issue we would like to comment on is the robustness of the characterizations (3.5) and (3.6) with respect to the independence
assumption for the components of the Lévy process Lt . First, note that if the components are dependent, then the key identity (3.1) is not any longer valid for ψk
defined as in (3.5) or (3.6). Let us determine how strong can it be violated. For
concreteness, assume that μl > 0 and that the dependence in the components of Lt
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33
34
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1
2
D. BELOMESTNY
is due to a correlation between diffusion components. In particular, let (k, l) > 0.
Since in the general case
3
7
and ∂uk uk ψ(u(k) ) = −σk2 − F[ν̄k ](u), we get
8
F[ν̄k ](u) + ψk (u) + σk2 =
9
10
11
12
13
14
15
5
6
7
(u(k) )
(u(k) )
8
φk
(k, l)
φk
u∂uk
+
.
(k)
2
φl (u )
φl (u(k) )
9
10
Using the fact that both functions u∂uk {φk (u(k) )/φl (u(k) )} and φk (u(k) )/φl (u(k) )
are uniformly bounded for u ∈ R, we get that the model “misspecification bias” is
bounded by C(k, l) with some constant C > 0. Thus, the weaker is the dependence between components Lk and Ll , the smaller is the resulting “misspecification bias.”
16
17
18
19
22
3.2. Algorithm. Set Zj = Yj − Y(j −1) , j = 1, . . . , n, and denote by
the
kth coordinate of Zj . Note that Zj , j = 1, . . . , n, are identically distributed. The
estimation procedure consists basically of three steps:
k (u) =
φ
(3.7)
25
26
27
(3.8)
29
(3.9)
31
n
1
Z k exp(iu Zj ),
n j =1 j
n
1
φlk (u) =
Z k Z l exp(iu Zj ),
n j =1 j j
28
30
n
1
llk (u) =
φ
Z k Z l Z l exp(iu Zj ).
n j =1 j j j
32
33
34
37
38
39
40
41
42
43
13
14
15
17
18
19
21
22
23
24
25
26
27
28
29
30
31
32
Step 2. In a second step, we estimate the second derivative of the characteristic
exponent ψk (u). Set
35
36
12
20
Step 1. First, we are interested in estimating partial derivatives of the function
φZ (u) up to the third order. To this end, define
23
24
11
16
Zjk
20
21
2
4
φl (u )
5
1
3
φk (u(k) )
∂uk ψ u(k) = ∂ul ψ u(k)
(k)
4
6
F:aos901.tex; (Laima) p. 10
(3.10)
(u(k) )φ
l (u(k) ) − φ
k (u(k) )φ
lk (u(k) )
φ
k,2 (u) = −1 φ
l (0) kk
ψ
,
l (u(k) )]2
[φ
√
l (0)| > κ/ n and
if |φ
(3.11)
k,2 (u) = −1 φ
ll (0)
ψ
33
34
35
36
37
38
kk (u(k) )φ
ll (u(k) ) − φ
k (u(k) )φ
llk (u(k) )
φ
otherwise, where κ is a positive number.
ll (u(k) )]2
[φ
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 11
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
Step 3. Finally, we construct an estimate for ν̄k (x) by applying the Fourier ink,2 (u):
version formula combined with a regularization to ψ
3
4
(3.12)
5
6
7
8
11
1
νk (x) = −
2π
11
12
13
R
where K(u) is a regularizing kernel supported on [−1, 1] and hn is a sequence of
bandwidths which tends to 0 as n → ∞. The choice of the sequence hn will be
discussed later on.
R EMARK 3.2. The parameter κ determines the testing error for the hypothesis
H : μl > 0. Indeed, if μl = 0, then φl (0) = 0 and by the central limit theorem
√ √
l (0)| > κ/ n ≤ P n|φ
l (0) − φl (0)| > κ
P |φ
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
→ P |ξ | > κ/ Var[Z l ] ,
15
17
4
5
6
7
8
9
14
16
2
3
k,2 (u) + σ 2 ]K(uhn ) du,
e−iux [ψ
k
9
10
1
n → ∞,
with ξ ∼ N (0, 1).
12
13
14
15
17
4.1. Global vs. local smoothness of Lévy densities. Let Lt be a one-dimensional Lévy process with a Lévy density ν. Denote ν̄(x) = x 2 ν(x). For any two
nonnegative numbers β and γ such that γ ∈ [0, 2] consider two following classes
of Lévy densities ν:
and
11
16
4. Asymptotic analysis. In this section, we are going to study the asymptotic
properties of the estimates νk (x), k = 1, . . . , d. In particular, we prove almost sure
uniform as well as pointwise convergence rates for νk (x). Moreover, we will show
the optimality of the above rates over suitable classes of time-changed Lévy models.
(4.1)
10
Sβ = ν :
R
(1 + |u|β )F[ν̄](u) du < ∞
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
()
ν(y) dy γ , → +0 ,
Bγ = ν :
|y|>
33
where is some positive function on R+ satisfying 0 < (+0) < ∞. The parameter γ is usually called the Blumenthal–Geetor index of Lt . This index γ is related
to the “degree of activity” of jumps of Lt . All Lévy measures put finite mass on
the set (−∞, −] ∪ [, ∞) for any arbitrary > 0. If ν([−, ]) < ∞ the process
has finite activity and γ = 0. If ν([−, ]) = ∞, that is, the process has infinite
activity and in addition the Lévy measure ν((−∞, −] ∪ [, ∞)) diverges near 0
at a rate ||−γ for some γ > 0, then the Blumenthal–Geetor index of Lt is equal
to γ . The higher γ gets, the more frequent the small jumps become.
35
(4.2)
34
36
37
38
39
40
41
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1
2
D. BELOMESTNY
Let us now investigate the connection between classes Sβ and Bγ . First, consider an example. Let Lt be a tempered stable Lévy process with a Lévy density
3
ν(x) =
4
5
6
ν̄(x) =
9
12
13
14
15
16
17
18
19
20
21
22
23
24
25
we derive
F[ν̄](u) =
0
8
9
eiux ν̄(x) dx 2γ γ (1 − γ )eiπ(1−γ /2) u−2+γ ,
u → +∞,
Hs (x0 , δ, D) = ν : ν̄(x) ∈ C (]x0 − δ, x0 + δ[),
s
(4.3)
sup
(l) ν̄ (x) ≤ D for l = 1, . . . , s .
x∈]x0 −δ,x0 +δ[
30
33
34
35
36
39
40
41
42
43
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
νk (x), we need the
4.2. Assumptions. In order to prove the convergence of assumptions listed below:
(AL1) The Lévy densities ν1 , . . . , νd are in the class Bγ for some γ > 0.
(AL2) For some p > 2, the Lévy densities νk , k = 1, . . . , d, have finite absolute
moments of the order p:
|x| νk (x) dx < ∞,
p
37
38
6
10
29
32
4
7
x
·γ
x 1−γ exp − 1(0,∞) (x),
(1 − γ )
2
28
31
2
5
by Erdélyi lemma [see Erdélyi (1956)]. Hence, ν cannot belong to Sβ as long as
β > 1 − γ . The message of this example is that given the activity index γ , the
parameter β determining the smoothness of ν̄, cannot be taken arbitrary large. The
above example can be straightforwardly generalized to a class of Lévy densities
supported on R+ . It turns out that if the Lévy density ν is supported on [0, ∞), is
infinitely smooth in (0, ∞) and ν ∈ Bγ for some γ ∈ (0, 1), then ν ∈ Sβ for all β
satisfying 0 ≤ β < 1 − γ and ν ∈
/ Sβ for β > 1 − γ . As a matter of fact, in the case
γ = 0 (finite activity case) the situation is different and β can be arbitrary large.
The above discussion indicates that in the case ν ∈ Bγ with some γ > 0 it is
reasonable to look at the local smoothness of ν̄k instead of the global one. To this
end, fix a point x0 ∈ R and a positive integer number s ≥ 1. For any δ > 0 and
D > 0 introduce a class Hs (x0 , δ, D) of Lévy densities ν defined as
26
27
∞
2γ
1
3
x > 0,
where γ ∈ (0, 1). It is clear that ν ∈ Bγ but what is about Sβ ? Since
8
11
x
·γ
x −(γ +1) exp − 1(0,∞) (x),
(1 − γ )
2
2γ
7
10
F:aos901.tex; (Laima) p. 12
R
k = 1, . . . , d.
32
33
34
35
36
37
38
(AT1) The time change T is independent of the Lévy process L and the sequence Tk = T (k) − T ((k − 1)), k ∈ N, is strictly stationary, α-mixing with
the mixing coefficients (αT (j ))j ∈N satisfying
αT (j ) ≤ ᾱ0 exp(−ᾱ1 j ),
31
j ∈ N,
39
40
41
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
for some positive constants ᾱ0 and ᾱ1 . Moreover, assume that
E[T −2/γ ()] < ∞,
2
3
4
5
8
9
Lt (z) = o(1),
K(u) = 1,
|z| → ∞,
Re z > 0.
u ∈ [−aK , aK ],
h−1
n
15
= O(n
1−δ
),
16
Mn
Mn = max
sup
22
23
24
25
26
27
28
29
n → ∞,
32
33
34
35
36
37
38
39
40
41
42
43
9
15
16
17
18
l
19
20
R EMARK 4.1. By requiring νk ∈ Bγ , k = 1, . . . , d, with some γ > 0, we exclude from our analysis pure compound Poisson processes and some infinite activity Lévy processes with γ = 0. This is mainly done for the sake of brevity: we
would like to avoid additional technical calculations related to the fact that the
distribution of Yt is not in general absolutely continuous in this case.
R EMARK 4.2. Assumption (AT1) is satisfied if, for example, the process T (t)
is of the form (1.1), where the rate process ρ(u) is strictly stationary, geometrically
α-mixing and fulfills
30
31
8
14
20
21
7
13
−1 (k) φ
u .
l
=k {|u|≤1/ hn }
19
6
12
for some positive number δ fulfilling 2/p < δ ≤ 1, where
18
5
11
log n 1
1
log
= o(1),
n
hn
hn
4
10
with some 0 < aK < 1.
(AH) The sequence of bandwidths hn is assumed to satisfy
14
17
3
(AK) The regularizing kernel K is uniformly bounded, is supported on [−1, 1]
and satisfies
11
13
2
E[T 2p ()] < ∞
Lt (z)/Lt (z) = O(1),
10
12
1
with γ and p being from the assumptions (AL1) and (AL2), respectively.
(AT2) The Laplace transform Lt (z) of T (t) fulfills
6
7
13
(4.4)
E[ρ 2p (u)] < ∞,
u ∈ [0, ],
E
−2/γ
ρ(u) du
0
< ∞.
In the case of the Cox–Ingersoll–Ross process ρ (see Section 5.2), the assumptions
(4.4) are satisfied for any p > 0 and any γ > 0.
R EMARK 4.3. Let us comment on the assumption (AH). Note that in order to
determine Mn , we do not need the characteristic function φ(u) itself, but only a low
bound for its tails. Such low bound can be constructed if, for example, a low bound
for the tail of Lt (z) and an upper bound for the Blumenthal–Geetor index γ are
available [see Belomestny (2010b) for further discussion]. In practice, of course,
one should prefer adaptive methods for choosing hn . One such method, based on
the so called “quasi-optimality” approach, is proposed and used in Section 5.1.
The theoretical analysis of this method is left for the future research.
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
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1
2
F:aos901.tex; (Laima) p. 14
D. BELOMESTNY
4.3. Uniform rates of convergence. Fix some k from the set {1, 2, . . . , d}. Define a weighting function w(x) = log−1/2 (e + |x|) and denote
3
4
x∈R
5
6
7
8
9
5
Let ξn be a sequence of positive r.v. and qn be a sequence of positive real
numbers. We shall write ξn = Oa.s. (qn ) if there is a constant D > 0 such that
P(lim supn→∞ qn−1 ξn ≤ D) = 1. In the case P(lim supn→∞ qn−1 ξn = 0) = 1, we
shall write ξn = oa.s. (qn ).
10
11
12
13
log3+ε n 1/ hn
β
2
νk L∞ (R,w) = Oa.s.
Rk (u) du + hn
ν̄k − 15
n
16
18
C OROLLARY 4.5.
and
24
|z| → ∞,
(4.5)
νk L∞ (R,w) = Oa.s.
ν̄k − 33
34
35
36
37
38
39
40
(4.6)
νk L∞ (R,w) = Oa.s.
ν̄k − 43
13
14
15
16
|L (z)| |z|−α ,
for some α > 0, then
νk L∞ (R,w) = Oa.s.
ν̄k − 22
23
24
Re z ≥ 0,
25
26
log
3+ε
n
n
β
exp(ac · h−η
n ) + hn
27
28
29
30
31
log3+ε n
η
β
exp(ac · h−γ
n ) + hn .
n
Choosing hn in such a way that the r.h.s. of (4.5) and (4.6) are minimized, we
obtain the rates shown in the Table 1. If γ ∈ (0, 1] in the assumption (AL1) and
41
42
12
21
with some constant c > 0. In the case μk = 0 we have
32
11
20
Suppose that σk = 0, γ ∈ (0, 1] in the assumption (AL1)
30
31
9
19
for some a > 0 and η > 0. If μk > 0, then
28
29
(1 + |ψk (u)|)2
.
|L (−ψk (u))|
|L (z)| exp(−a|z|η ),
25
27
8
18
21
26
7
17
Rk (u) =
20
23
−1/ hn
for arbitrary small ε > 0, where
19
22
6
10
T HEOREM 4.4. Suppose that the assumptions (AL1), (AL2), (AT1), (AT2),
(AK) and (AH) are fulfilled. Let νk (x) be the estimate for ν̄k (x) defined in Section 3.2. If νk ∈ Sβ for some β > 0, then
14
17
2
3
ν̄k − νk L∞ (R,w) = sup [w(|x|)|ν̄k (x) − νk (x)|].
4
1
|z| → ∞,
Re z ≥ 0,
32
33
34
35
36
37
38
39
log
3+ε
n
n
hn−1/2−α + hβn
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 15
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
TABLE 1
νk in the case σk = 0
Uniform convergence rates for 1
2
3
|L (z)| |z|−α
4
5
μk > 0
6
n−β/(2α+2β+1)
× log(3+ε)β/(2α+2β+1) (n)
7
8
15
1
2
|L (z)| exp(−a|z|η )
μk = 0
μk > 0
μk = 0
n−β/(2αγ +2β+1)
× log(3+ε)β/(2αγ +2β+1) (n)
log−β/η n
log−β/γ η n
11
14
15
16
17
18
19
20
23
24
25
26
27
28
29
30
31
32
35
νk L∞ (R,w) = Oa.s.
ν̄k − log3+ε n −1/2−αγ
hn
+ hβn .
n
The choices hn = n−1/(2(α+β)+1) log(3+ε)/(2(α+β)+1) (n) and
38
41
42
43
13
14
15
16
17
for the cases μk > 0 and μk = 0, respectively, lead to the bounds shown in Table 1.
In the case σk > 0, the rates of convergence are given in Table 2.
19
18
20
21
R EMARK 4.6. As one can see, the assumption (AH) is always fulfilled for
the optimal choices of hn given in Corollary 4.5, provided αγ + β > 0 and p >
2 + 1/(αγ + β).
4.4. Pointwise rates of convergence. Since the transformed Lévy density ν̄k is
usually not smooth at 0 (see Section 4.1), pointwise rates of convergence might
be more informative than the uniform ones if νk ∈ Bγ for some γ > 0. It is remarkable that the same estimate νk as before will achieve the optimal pointwise
convergence rates in the class Hs (x0 , δ, D), provided the kernel K satisfies (AK)
and is sufficiently smooth.
22
23
24
25
26
27
28
29
30
31
32
33
T HEOREM 4.7. Suppose that the assumptions (AL1), (AL2), (AT1), (AT2),
(AK) and (AH) are fulfilled. If νk ∈ Hs (x0 , δ, D) with Hs (x0 , δ, D) being defined
34
35
36
TABLE 2
νk in the case σk > 0
Uniform convergence rates for 39
40
12
hn = n−1/(2(αγ +β)+1) log(3+ε)/(2(αγ +β)+1) (n)
36
37
7
11
33
34
6
10
provided μk > 0. In the case μk = 0, one has
21
22
5
9
12
13
4
8
9
10
3
37
38
39
|L (z)| |z|−α
|L (z)| exp(−a|z|η )
n−β/(4α+2β+1) log(3+ε)β/(4α+2β+1) (n)
log−β/2η n
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
16
1
D. BELOMESTNY
in (4.3), for some s ≥ 1, δ > 0, D > 0, and K ∈ C m (R) for some m ≥ s, then
2
3
4
(4.7)
log3+ε n 1/ hn
|
νk (x0 ) − ν̄k (x0 )| = Oa.s.
R2k (u) du + hsn
n
5
6
7
8
F:aos901.tex; (Laima) p. 16
−1/ hn
11
12
13
14
15
with Rk (u) as in Theorem 4.4. As a result, the pointwise rates of convergence
for different asymptotic behaviors of the Laplace transform Lt coincide with ones
given in Tables 1 and 2, if we replace β with s.
18
19
20
21
R EMARK 4.8. If the kernel K is infinitely smooth, then it will automatically
“adapt” to the pointwise smoothness of ν̄k , that is, (4.7) will hold for arbitrary large
s ≥ 1, provided νk ∈ Hs (x0 , δ, D) with some δ > 0 and D > 0. An example of
infinitely smooth kernels satisfying (AK) is given by the so called flat-top kernels
(see Section 5.1 for the definition).
24
25
26
27
T HEOREM 4.9. Let Lt be a Lévy process with zero diffusion part, a drift μ
and a Lévy density ν. Consider a time-changed Lévy process Yt = LT (t) , where
the Laplace transform of the time change T (t) fulfills
(4.8)
(k+1)
(k)
L (z)/L (z) = O(1),
32
33
34
35
36
37
38
39
40
41
42
43
|z| → ∞,
Re z ≥ 0,
7
8
10
11
12
13
14
15
17
18
19
20
21
(4.9)
lim inf inf
sup
n→∞ ν ν∈S ∩B
γ
β
n
25
26
29
P(ν,T ) ν̄ − ν L∞ (R,w) > εhβn log−1 (1/ hn ) > 0
2 2β+1
[L (c · h−γ
n )] hn
24
28
30
31
32
33
for any ε > 0 and any sequence hn satisfying
−1
23
27
for k = 0, 1, 2, and uniformly in ∈ [0, 1]. Then
30
31
6
22
28
29
4
16
4.5. Lower bounds. In this section, we derive a lower bound on the minimax
risk of an estimate ν (x) over a class of one-dimensional time-changed Lévy processes Yt = LT (t) with the known distribution of T , such that the Lévy measure
ν of the Lévy process Lt belongs to the class Sβ ∩ Bγ with some β > 0 and
γ ∈ (0, 1]. The following theorem holds
22
23
3
9
16
17
2
5
9
10
1
34
= O(1),
n → ∞,
36
in the case μ = 0 and
2 2β+1
n−1 [L (c · h−1
= O(1),
n )] hn
35
37
n → ∞,
in the case μ > 0, with some positive constant c > 0. Note that the infimum in (4.9)
is taken over all estimators of ν based on n observations of the r.v. Y and P(ν,T )
stands for the distribution of n copies of Y .
38
39
40
41
42
43
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17
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
C OROLLARY 4.10. Suppose that the underlying Lévy process is driftless, that
is, μ = 0 and Lt (z) = exp(−azt) for some a > 0, corresponding to a deterministic
time change process T (t) = at. Then by taking
4
log n − ((2β + 1)/γ ) log log n
hn =
2ac
5
6
7
8
9
we arrive at
−1/γ
12
13
6
7
lim inf inf
sup
n→∞ ν ν∈S ∩B
γ
β
P(ν,T ) ν̄ − ν L∞ (R,w) > ε · β/γ log−β/γ n > 0.
20
21
22
23
24
25
11
hn = (n)−1/(2αγ +2β+1)
14
28
29
32
33
34
35
36
37
38
39
40
41
42
43
13
16
we get
lim inf inf
sup
n→∞ ν ν∈S ∩B
γ
β
P(ν,T ) ν̄ − ν L∞,w (R) > ε · (n)−β/(2αγ +2β+1) log−1 n > 0,
17
18
19
where α = α0 + 1.
20
R EMARK 4.12. The Theorem 4.9 continues to hold for → 0 and therefore
can be used to derive minimax lower bounds for the risk of ν in high-frequency
setup. As can be seen from Corollaries 4.10 and 4.11, the rates will strongly depend
on the specification of the time change process T .
21
22
23
24
25
26
The pointwise rates of convergence obtained in (4.7) turn out to be optimal over
the class Hs (x0 , δ, D) ∩ Bγ with s ≥ 1, δ > 0, x0 ∈ R, D > 0 and γ ∈ (0, 1] as the
next theorem shows.
30
31
12
15
26
27
9
C OROLLARY 4.11. Again let μ = 0. Take Lt
> 0 for
some α0 > 0, resulting in a Gamma process T (t) (see Section 5.1 for the definition). Under the choice
17
19
8
10
(z) = 1/(1 + z)α0 t , Re z
15
18
3
5
,
14
16
2
4
10
11
1
27
28
29
30
T HEOREM 4.13. Let Lt be a Lévy process with zero diffusion part, a drift μ
and a Lévy density ν. Consider a time-changed Lévy process Yt = LT (t) , where
the Laplace transform of the time change T (t) fulfills (4.8). Then
(4.10) lim inf inf
sup
n→∞ ν ν∈H (x ,δ,D)∩B
s 0
γ
P(ν,T ) |ν̄(x0 ) − ν (x0 )| > εhsn log−1 (1/ hn ) > 0
for s ≥ 1, δ > 0, D > 0, any ε > 0 and any sequence hn satisfying
2 2s+1
= O(1),
n−1 [L (c · h−γ
n )] hn
n → ∞,
in the case μ > 0, with some positive constant c > 0.
32
33
34
35
36
37
38
39
in the case μ = 0 and
2 2s+1
= O(1),
n−1 [L (c · h−1
n )] hn
31
40
n → ∞,
41
42
43
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1
F:aos901.tex; (Laima) p. 18
D. BELOMESTNY
4.6. Extensions.
1
2
3
4
5
6
7
8
2
One-dimensional time-changed Lévy models. Let us consider a class of onedimensional time-changed Lévy models (2.7) with the known time change process, that is, the known function Lt for all t > 0. This class of models trivially
includes Lévy processes without time change [by setting Lt (z) = exp(−tz)] studied in Neumann and Reiß (2009) and Comte and Genon-Catalot (2010). We have
in this case
9
10
(4.11)
φ
ψ1 (u) = −
11
12
(u)L (−ψ (u)) − φ (u)L (−ψ (u))/L (−ψ (u))
1
1
1
2
[L (−ψ1 (u))]
with
17
18
19
20
23
24
25
30
31
The case of the unknown (σk ). One way to proceed in the case of the unknown
νk (x) = x 4 νk (x). Assuming νk (x) dx <
(σk ) and νk ∈ Bγ with γ < 2 is to define ∞, we get
ψk(4) (u) =
R
e
iux
νk (x) dx.
Estimation of L . Let us first estimate ψk . Set
k (u) = −1 φ
l (0)
ψ
17
18
19
20
k L (R,w) = Oa.s.
ψk − ψ
∞
log
3+ε
n
25
28
29
30
33
34
35
36
37
39
with a weighting function
w(u) =
24
38
n
39
41
23
32
u (k)
φk (v )
dv.
l (v (k) )
0 φ
(4.12)
22
31
Under Assumptions (AL2), (AT1), (AT2), (AK) and (AH) we derive
37
43
16
27
36
42
11
26
Hence, in the above situation one can apply the regularized Fourier inversion for(4)
mula to an estimate of ψk (u) instead of ψk (u).
34
40
10
21
33
38
9
15
32
35
8
is an inverse function for L . Thus, ψ1 (u) is again a ratio-type estimate
involving the derivatives of the c.f. φ up to second order, that agrees with the one
proposed in Comte and Genon-Catalot (2010) for the case of pure Lévy processes.
Although we do not study the case of one-dimensional models in this work, our
analysis can be easily adapted to this situation as well. In particular, the derivation
of the pointwise convergence rates can be directly carried over to this situation.
27
29
7
14
where L−
26
28
6
13
21
22
5
ψ1 (u) = −L−
(φ(u)),
14
16
4
12
13
15
3
40
u
1 + |ψk (v)|
0
|L (−ψk (v))|
−1
dv
41
.
42
43
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
4
5
6
7
Now let us define an estimate for L as a solution of the following optimization
problem
10
11
= arg inf sup w(u)L(−ψ
k (u)) − φ
u(k) ,
L
(4.13)
L∈M u∈R
(4.14)
(−ψk (u)) − L (−ψk (u))|} = Oa.s.
sup{w(u)|L
u∈R
14
15
16
17
18
19
20
21
22
23
24
n
L(u) =
∞
0
n
.
e−ux dF (x)
with some distribution function F satisfying x dF (x) = , we can replace the
optimization over M in (4.13) by the optimization over the corresponding set of
distribution functions. The advantage of the latter approach is that herewith we can
directly get an estimate for the distribution function of the r.v. T (). A practical
implementation of the estimate (4.13) is still to be worked out, as the optimization
over the set M is not feasible and should be replaced by the optimization over
suitable approximation classes (sieves). Moreover, the “optimal” weights in (4.13)
depend on the unknown L. However, it turns out that it is possible to use any
weighting function which is dominated by w(u), that is, one needs only some
lower bounds for L .
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
2
4
Since any function L from M has a representation
12
13
log
3+ε
1
3
where M is the set of completely monotone functions L satisfying L(0) = 1 and
L (0) = −. Simple calculations and the bound (4.12) yield
8
9
19
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
R EMARK 4.14. It is interesting to compare (4.12) and (4.14) with Theorem 3.2 in Horowitz and Mammen (2007). At first sight it may seem strange that,
while the rates of convergence for our “link” function L and the “components”
ψk depend on the tail behavior of L , the rates in Horowitz and Mammen (2007)
rely only on the smoothness of the link function and the components. The main
reason for this is that the derivative of the link function in the above paper is assumed to be uniformly bounded from below [assumption (A8)], a restriction that
can be hardly justifiable in our setting. The convergence analysis in the unbounded
case is, in our opinion, an important contribution of this paper to the problem of
estimating composite functions that can be carried over to other setups and settings.
4.7. Discussion. As can be seen, the estimate νk can exhibit various asymptotic behavior depending on the underlying Lévy process Lt and the time-change
T (t). In particular, if the Laplace transform Lt (z) of T dies off at exponential rate
as Re z → +∞ and μk = 0, then the rates of convergence of νk are logarithmic
and depend on the Blumenthal–Geetor index of the Lévy process Lt . The larger
is the Blumenthal–Geetor index, the slower are the rates and the more difficult
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
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1
2
3
4
5
6
7
F:aos901.tex; (Laima) p. 20
D. BELOMESTNY
the estimation problem becomes. For the polynomially decaying Lt (z) one gets
polynomial convergence rates that also depend on the Blumenthal–Geetor index
of Lt . Let us also note that the uniform rates of convergence are usually rather
slow, since β < 1 − γ in most situations. The pointwise convergence rates for
points x0 = 0 can, on the contrary, be very fast. The rates obtained turn out to be
optimal up to a logarithmic factor in the minimax sense over the classes Sβ ∩ Bγ
and Hs (x0 , δ, D) ∩ Bγ .
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
5. Simulation study. In our simulation study, we consider two models based
on time-changed normal inverse Gaussian (NIG) Lévy processes. The NIG Lévy
processes is a relatively new class of processes introduced in Barndorff-Nielsen
(1998) as a model for log returns of stock prices. The processes of this type
are characterized by the property that their increments have NIG distribution.
Barndorff-Nielsen (1998) considered classes of normal variance–mean mixtures
and defined the NIG distribution as the case when the mixing distribution is inverse Gaussian. Shortly after its introduction, it was shown that the NIG distribution fits very well the log returns on German stock market data, making the NIG
Lévy processes of great interest for practioneers. A NIG distribution has in general
four parameters: α ∈ R+ , κ ∈ R, δ ∈ R+ and μ ∈ R with |κ| < α. Each parameter
in NIG(α, κ, δ, μ) distribution can be interpreted as having a different effect on
the shape of the distribution: α is responsible for the tail heaviness of steepness,
κ has to do with symmetry, δ scales the distribution and μ determines its mean
value. The NIG distribution is infinitely divisible with c.f.
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
φ(u) = exp δ
25
27
2
3
4
5
6
7
8
24
26
1
α 2 − κ 2 − α 2 − (κ + iu)2 + iμu .
Therefore, one can define the NIG Lévy process (Lt )t≥0 which starts at zero and
has independent and stationary increments such that each increment Lt+ − Lt
has NIG(α, κ, δ, μ) distribution. The NIG process has no diffusion component
making it a pure jump process with the Lévy density
2αδ exp(κx)K1 (α|x|)
,
ν(x) =
π
|x|
(5.1)
K1 (z) 2/z,
z → +0,
and
K1 (z) π −z
e ,
2z
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
32
z → +∞,
we conclude that ν ∈ B1 and ν ∈ Hs (x0 , δ, D) for arbitrary large s > 0 and some
δ > 0, D > 0, if x0 = 0. Moreover, the assumption (AL2) is fulfilled for any p > 0.
Furthermore, the identity
d2
2 2
2 3/2
log
φ(u)
=
−α
/
α
−
(κ
+
iu)
du2
10
31
where Kλ (z) is the modified Bessel function of the third kind. Taking into account
the asymptotic relations
9
33
34
35
36
37
38
39
40
41
42
43
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
21
implies ν ∈ S2−δ for arbitrary small δ > 0. In the next sections are going to study
two time-changed NIG processes: one uses the Gamma process as a time change
and another employs the integrated CIR processes to model T .
4
5
6
7
νT (x) = θ x −1 exp(−λx),
9
11
−θ t
Lt (z) = (1 + z/λ)
13
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
34
35
36
37
38
39
40
41
42
43
K(x) =
⎧
1,
⎪
⎪
⎨
⎪ exp −
⎪
⎩
0,
e−1/(|x|−0.05)
1 − |x|
11
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
33
34
|x| ≥ 1.
35
The flat-top kernels obviously satisfy the assumption (AK). Thus, all assumptions
of Theorem 4.4 are fulfilled and Corollary 4.5 leads to the following convergence
ν1 of the function ν̄1 (x) = x 2 ν(x):
rates for the estimate ν1 L∞ (R,w) = Oa.s.
ν̄1 − 10
32
0.05 < |x| < 1,
−(1−δ )/(θ +5/2)
n
log(3+ )/(θ +5/2) (n) ,
7
13
|x| ≤ 0.05,
,
6
12
Re z ≥ 0.
,
5
9
It follows from the properties of the Gamma and the corresponding inverse Gamma
distributions that the assumptions (AT1) and (AT2) are fulfilled for the Gamma
process T , provided θ > 2/γ . Consider now the time-changed Lévy process
Yt = LT (t) where Lt = (L1t , L2t , L3t ) is a three-dimensional Lévy process with independent NIG components and T is a Gamma process. Note that the process Yt
is a multidimensional Lévy process since T was itself the Lévy process. Let us be
more specific and take the -increments of the Lévy processes L1t , L2t and L3t
to have NIG(1, −0.05, 1, −0.5), NIG(3, −0.05, 1, −1) and NIG(1, −0.03, 1, 2)
distributions, respectively. Take also θ = 1 and λ = 1 for the parameters of the
Gamma process T . Next, fix an equidistant grid on [0, 10] of the length n = 1,000
and simulate a discretized trajectory of the process Yt . Let us stress that the dependence structure between the components of Yt is rather flexible (although they
are uncorrelated) and can be efficiently controlled by the parameters of the corν1 as described in
responding Gamma process T . Next, we construct an estimate Section 3.2. We first estimate the derivatives φ1 , φ2 , φ11 and φ12 by means of (3.7)
and (3.8). Then we estimate ψ1 (u) using the formula (3.10) with k = 1 and l = 2.
ν1 from (3.12) where the kernel K is chosen to be the so called
Finally, we get flat-top kernel of the form
32
33
3
8
x ≥ 0,
where the parameter θ controls the rate of jump arrivals and the scaling parameter
λ inversely controls the jump size. The Laplace transform of T is of the form
12
14
2
4
5.1. Time change via a Gamma process. Gamma process is a Lévy process
such that its increments have Gamma distribution, so that T is a pure-jump increasing Lévy process with the Lévy density
8
10
1
n → ∞,
with arbitrary small positive numbers δ and , provided the sequence hn is chosen
ν1 .
as in Corollary 4.5. Let us turn to the finite sample performance of the estimate 36
37
38
39
40
41
42
43
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F:aos901.tex; (Laima) p. 22
D. BELOMESTNY
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
15
16
F IG . 1. Left-hand side: objective function f (l) for “quasi-optimality” approach versus the correν1 (dashed line) together
sponding bandwidths hl , l = 1, . . . , 40. Right-hand side: adaptive estimate with the true function ν̄1 (solid line).
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
38
39
40
41
42
43
15
16
17
It turns out that the choice of the sequence hn is crucial for a good performance
of ν1 . For this choice, we adopt the so called “quasi-optimality” approach proposed
in Bauer and Reiß (2008). This approach is aimed to perform a model selection in
inverse problems without taking into account the noise level. Although one can
prove the optimality of this criterion on average only, it leads in many situations
to quite reasonable results. In order to implement the “quasi-optimality” algorithm
in our situation, we first fix a sequence of bandwidths h1 , . . . , hL and construct
(1)
(L)
the estimates ν1 , . . . , ν1 using the formula (3.12) with bandwidths h1 , . . . , hL ,
respectively. Then one finds l = arg minl f (l) with
" (l+1)
f (l) = "
ν1
"
(l)
−
ν1 "L1 (R) ,
l = 1, . . . , L.
18
19
20
21
22
23
24
25
26
27
28
29
∗
Denote by ν1 = ν1l a new adaptive estimate for ν̄1 . In our implementation of the
“quasi-optimality” approach, we take hl = 0.5 + 0.1 × l, l = 1, . . . , 40. In Figure 1,
the sequence f (l), l = 1, . . . , 40, is plotted. On the right-hand side of Figure 1,
we show the resulting estimate ν1 together with the true function ν̄1 . Based on
the
estimate
ν
,
one
can
estimate
some functionals of ν̄1 . For example, we have
1
ν1 (x) dx = 1.049053 [ ν̄1 (x) dx = 1.015189].
36
37
14
30
31
32
33
34
35
36
5.2. Time change via an integrated CIR process. Another possibility to construct a time-changed Lévy process from the NIG Lévy process Lt is to use a time
change of the form (2.2) with some rate process ρ(t). A possible candidate for the
rate of the time change is given by the Cox–Ingersoll–Ross process (CIR process).
The CIR process is defined as a solution of the following SDE:
#
dZt = κ(η − Zt ) dt + ζ Zt dWt ,
Z0 = 1,
37
38
39
40
41
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STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
4
5
where Wt is a Wiener process. This process is mean reverting with κ > 0 being the
speed of mean reversion, η > 0 being the long-run mean rate and ζ > 0 controlling
the volatility of Zt . Additionally, if 2κη > ζ 2 and Z0 has Gamma distribution, then
Zt is stationary and exponentially α-mixing [see, e.g., Masuda (2007)]. The time
change T is then defined as
6
T (t) =
7
8
9
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
t
0
Lt (z) =
(cosh(γ (z)t/2) + κ sinh(γ (z)t/2)/γ (z))2κη/ζ
2
√
with γ (z) = κ 2 + 2ζ 2 z. It is easy to see that Lt (z) exp(− ζ2z [1 + tκη]) as
|z| → ∞ with Re z ≥ 0. Moreover, it can be shown that E|T (t)|p < ∞ for any
p ∈ R. Let Lt be again a three-dimensional NIG Lévy process with independent
components distributed as in Section 5.1. Construct the time-changed process Yt =
LT (t) . Note that the process Yt is not any longer a Lévy process and has in general
dependent increments. Let us estimate ν̄1 , the transformed Lévy density of the
ν1
first component of Lt . First, note that according to Theorem 4.4, the estimate constructed as described in Section 3.2, has the following logarithmic convergence
rates
ν̄1 − ν1 L∞ (R,w) = Oa.s. log
2
3
4
5
7
exp(κ 2 ηt/ζ 2 ) exp(−2z/(κ + γ (z) coth(γ (z)t/2)))
1
6
Zt dt.
Simple calculations show that the Laplace transform of T (t) is given by
10
11
23
−2(2−δ)
(n) ,
n → ∞,
for arbitrary small δ > 0, provided the bandwidth sequence is chosen in the opν1 with the choice of hn based on
timal way. Finite sample performance of the “quasi-optimality” approach is illustrated in Figure 2 where the sequence
of estimates ν1(1) , . . . , ν1(L) was constructed from the time series Y , . . . , Yn
with n = 5,000 and = 0.1. The parameters of the used CIR process are
ν1 . We
κ = 1, η = 1 and ζ = 0.1. Again we can compute some functionals of estimates for the integral
have, for example, following
and for the mean of ν̄1 :
ν
(x)
dx
=
1.081376
[
ν̄
(x)
dx
=
1.015189]
and
x
ν1 (x) dx = −0.4772505
1
1
[ x ν̄1 (x) dx = −0.3057733].
Let us now test the performance of estimation algorithm in the case of a timechanged NIG process (parameters are the same as before), where the time change
is again given by the integrated CIR process with the parameters η = 1, ζ = 0.1
and κ ∈ {0.05, 0.1, 0.5, 1}. Figure 3(left) shows the boxplots of the resulting error
ν1 L∞ (R,w) computed using 100 trajectories each of the length n = 5,000,
ν̄1 − where the time span between observation is = 0.1. Note that if our time units
are days, then we get about 2 years of observations with about one mean reversion
per month in the case κ = 0.05. As one can see, the performance of the algorithm
remains reasonable for the whole range of κ. In Figure 3(right), we present the
ν1 L∞ (R,w) in the case of η = 1, ζ = 0.1, κ = 1 and n ∈
boxplots of the error ν̄1 − 8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
24
F:aos901.tex; (Laima) p. 24
D. BELOMESTNY
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
15
16
F IG . 2. Left-hand side: objective function f (l) for the “quasi-optimality” approach versus the
ν1 (dashed line) together with the
corresponding bandwidths hl . Right-hand side: adaptive estimate true function ν̄1 (solid line).
17
18
19
20
{500, 1,000, 3,000, 5,000}. As one can expect, the performance of the algorithm
becomes worse as n decreases. However, the quality of the estimation remains
reasonable even for n = 500.
25
26
27
28
29
30
31
6.1. Proof of Theorem 4.4. For simplicity, let consider the case of μl > 0 and
σk = 0. By Proposition 7.4 [take Gn (u, z) = exp(iuz), Ln = μ̄n = σ̄n = 1, a =
0, b = 1]
√ l (0)| ≤ κ/ n ≥ P |φ
l (0) − φl (0)| > μl ≤ Bn−1−δ
P |φ
for some constants δ > 0, B > 0 and n large enough. Furthermore, simple calculations lead to the following representation:
k,2 (u) =
ψk (u) − ψ
(6.1)
38
39
40
41
42
43
ψk (u) l (0)
φl (0) − φ
ψl (0)
20
24
25
26
27
28
29
30
31
32
33
34
+ R0 (u) + R1 (u) + R2 (u),
35
37
19
23
33
36
18
22
6. Proofs of the main results.
32
34
16
21
23
24
15
17
21
22
14
35
36
where
l u(k)
R0 (u) = [V1 (u)ψk (u) − V2 (u)ψk (u)] φl u(k) − φ
k u(k)
+ V2 (u) φk u(k) − φ
kk u(k)
− V1 (u) φkk u(k) − φ
lk u(k) ,
+ V1 (u)ψk (u) φlk u(k) − φ
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 25
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
25
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
15
16
16
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
31
31
32
32
33
33
34
34
35
35
36
36
37
37
38
38
39
39
40
41
42
43
40
Boxplots of the error ν̄1 − ν1 L∞ (R,w) for different values of the mean reversion speed
F IG . 3.
parameter κ and different numbers of observations n.
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
26
D. BELOMESTNY
2
5
8
9
11
13
V1 (u) =
17
18
19
21
7
8
9
10
11
12
13
15
16
20
and
V
2 (u) = 2 (u) − 1 V2 (u)
−1
1 (k) (k) φl u
− φl u
.
φl (u(k) )
The representation (6.1) and the Fourier inversion formula imply the following
representation for the deviation ν̄k − νk :
(u) = 1 −
l (0))
1 (φl (0) − φ
ν̄k (x) − νk (x) =
2π
ψl (0)
R
e
−iux
ψk (u)K(uhn ) du
1
e−iux R0 (u)K(uhn ) du
2π R
1
e−iux R1 (u)K(uhn ) du
+
2π R
1
e−iux R2 (u)K(uhn ) du
+
2π R
1
+
e−iux 1 − K(uhn ) ψk (u) du.
2π R
+
33
34
35
36
37
38
43
6
V
1 (u) = (u) − 1 V1 (u),
32
42
5
17
31
41
L (−ψk (u))
φl (0)φlk (u(k) )
,
V2 (u) =
=
−V
(u)ψ
(u)
1
k
[φl (u(k) )]2
L (−ψk (u))
30
40
φl (0)
1
=− ,
(k)
φl (u )
L (−ψk (u))
20
39
4
14
16
29
3
with
15
28
1
2
(φl (0) − φl (0)) R0 + R1
+
φl (u(k) )
φl (0)
12
27
l u(k) ψ (u) − φk u(k) − φ
k u(k)
× φl u(k) − φ
k
10
26
lk (u(k) ))
φl (0)(φlk (u(k) ) − φ
R2 (u) = 2 (u)
[φl (u(k) )]2
7
25
lk u(k) ,
+ V
1 (u)ψk (u) φlk u(k) − φ
6
24
kk u(k)
− V
1 (u) φkk u(k) − φ
4
23
k u(k)
+ V
2 (u) φk u(k) − φ
3
22
1 (u)ψ (u) − V
2 (u)ψ (u)] φl u(k) − φ
l u(k)
R1 (u) = [V
k
k
1
14
F:aos901.tex; (Laima) p. 26
First, let us show that
log3+ε n 1/ hn
−iux
2
R1 (u)K(uhn ) du = oa.s
Rk (u) du
sup e
n
x∈R
R
−1/ hn
18
19
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 27
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
27
and
1
3+ε log
n
R2k (u) du .
sup e−iux R2 (u)K(uhn ) du = oa.s
n
R
x∈R R
2
3
4
5
6
9
10
|u|≤1/ hn
11
12
×
13
14
1/ hn
−1/ hn
u∈R
A=
18
%
(k) log n
w(|u|)
φl u
− φl u(k) ≤ ξ
.
n
{|u|≤1/ hn }
23
|u|<1/ hn
l
24
=o
25
#
n → ∞,
hn ,
26
28
29
30
31
32
33
34
37
38
39
40
41
42
43
16
17
18
19
21
22
23
24
25
26
and hence
sup
(6.2)
{|u|≤1/ hn }
|1 − (u)| = o
#
27
n → ∞.
hn ,
=o
−1/ hn
2 1/ hn
log3+ε n 1/ hn
hn log n
Rk (u) du = o
R2k (u) du
n
29
31
1/ h
n
l u(k) K(uhn ) du
sup e−iux (u) − 1 V1 (u)ψk (u) φl u(k) − φ
x∈R
28
30
Therefore, one has on A
35
36
15
20
l (u(k) ) φl (u(k) ) − φ
≤ ξ Mn w −1 (1/ hn ) log n/n
sup φ (u(k) )
22
10
13
By the assumption (AH), it holds on A
21
9
14
sup
19
8
12
|V1 (u)||ψk (u)| du
$
17
7
11
with w(u) = log−1/2 (e + u), u ≥ 0. Fix some ξ > 0 and consider the event
16
27
4
6
e−iuz (u) − 1 V1 (u)ψ (u) φl u(k) − φ
l u(k) K(uhn ) du
k
R
l u(k) w −1 (1/ hn )
≤ sup |(u) − 1| sup w(|u|)φl u(k) − φ
8
20
3
5
We have, for example, for the first term in R1 (u)
7
15
2
n
−1/ hn
−1/ hn
since ψk (u) and K(u) are uniformly bounded on R. On the other hand, Proposition 7.4 implies [on can take Gn (u, z) = exp(iuz), Ln = μ̄n = σ̄n = 1, a = 0,
b = 1]
P(Ā) n
−1−δ ,
n → ∞,
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
28
1
D. BELOMESTNY
for some δ > 0. The Borel–Cantelli lemma yields
3
x∈R
4
5
6
10
1/ hn
12
−1/ hn
(6.3)
15
hn
Kn (z) =
1
−1
35
36
37
38
43
20
sup {pk,t (x)} t
(6.4)
21
22
−1/γ
,
24
25
t → 0,
−1/γ
,
t → 0.
33
34
Moreover, under assumption (AL2) [see Luschgy and Pagès (2008)]
(6.6)
and
|x|m pk,t (x) dx = O(t),
(6.7)
29
32
x∈R
28
31
sup {pl,t (x)} t
(6.5)
27
30
x∈R
|x|m pl,t (x) dx = O(t),
35
t → 0,
36
37
38
40
42
19
26
39
41
e
V1 (u/ hn )ψk (u/ hn )K(u) du.
Since νk , νl ∈ Bγ for some γ > 0 [assumption (AL1)], the Lévy processes Lkt and
Llt possess infinitely smooth densities pk,t and pl,t which are bounded for t > 0
[see Sato (1999), Section 28] and fulfill [see Picard (1997)]
33
34
15
23
31
32
14
18
−iuz
25
30
hn
13
1
u−z
Gn (u, z) = Kn
.
hn
hn
24
29
12
Now we are going to make use of Proposition 7.4 to estimate the term S (x) on the
r.h.s. of (6.3). To this end, let
23
28
10
17
20
27
hn
9
11
with
19
26
8
16
18
22
4
7
l u(k) K(uhn ) du
e−iux V1 (u)ψk (u) φl u(k) − φ
nhn j =1
16
21
3
6
−1/ hn
2
5
x − Zk n x − Zk
1 j
l
l 1
Z j Kn
−E Z
Kn
= S (x)
=
14
17
log3+ε n 1/ hn
2
= oa.s.
Rk (u) du .
Other terms in R1 and R2 can be analyzed in a similar way. Turn now to the rate
determining term R0 . Consider, for instance, the integral
11
13
−1/ hn
n
7
9
1
1/ h
n
(k) (k) −iux e
(u) − 1 V1 (u)ψk (u) φl u
− φl u
K(uhn ) du
sup 2
8
F:aos901.tex; (Laima) p. 28
39
|x| pk,t (x) dx = O(t ),
m
m
40
41
|x| pl,t (x) dx = O(t ),
m
m
t → +∞,
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 29
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
for any 2 ≤ m ≤ p. As a result, the distribution of (Z k , Z l ) is absolutely continuous
with uniformly bounded density qkl given by
3
qkl (y, z) =
4
5
6
7
8
9
10
1
E[|Z l |2 |Gn (u, Z k )|2 ] = 2
hn
C0
≤
hn
12
13
≤ C1
14
16
∞
0
R
|Kn (v)| dv
−1/ hn
E[|Z k |4 |Gn (u, Z k )|2 ] ≤ C3
21
23
24
25
E[|Z k |2 |Z l |2 |Gn (u, Z k )|2 ] ≤ C4
σ̄n2 = C
27
1/ hn
−1/ hn
μ̄n = K∞ ψ ∞
29
30
31
Ln = K∞ ψ ∞
32
33
34
35
36
37
38
39
40
41
42
43
15
1/ hn
−1/ hn
1/ hn
−1/ hn
1/ hn
−1/ hn
16
17
|V1 (u)|2 du,
18
19
|V1 (u)|2 du,
20
21
22
|V1 (u)|2 du
23
24
25
26
1/ hn
−1/ hn
1/ hn
−1/ hn
27
28
|V1 (u)| du,
29
30
31
|u||V1 (u)| du,
32
where C = maxk=1,2,3,4 {Ck }. Since |V1 (u)| → ∞ as |u| → ∞ and hn → ∞, we
get μ̄n /σ̄n2 = O(1). Furthermore, due to assumption (AH)
3/2
(6.8) μ̄n hn−1/2 σ̄n n1/2−δ/2 σ̄n ,
Ln h3/2
σ̄n ,
n σ̄n n
−1/2
and σ̄n = O(hn Mn ) = O(n1/2 ). Thus, the assumptions (AG1)
n → ∞,
and (AG2) of
Proposition 7.4 are fulfilled. The assumption (AZ1) follows from Lemma 7.1 and
the assumption (AT1). Therefore, we get by Proposition 7.4
P sup[w(|z|)|S (z)|] ≥ ξ
z∈R
9
14
|V1 (u)|2 du,
28
8
13
with some positive constants C2 , C3 and C4 . Define
26
7
11
|V1 (u)|2 du
19
20
6
12
1/ hn
E[|Z k |2 |Gn (u, Z k )|2 ] ≤ C2
18
5
10
with some finite constants C0 > 0 and C1 > 0. Similarly,
17
22
R
n
2
R
2
4
u − y 2
2
Kn
|z|
q
(y,
z)
dz
dy
kl
h
1
3
pk,t (y)pl,t (z) dπ(dt),
where π is the distribution function of the r.v. T (). The asymptotic relations
(6.4)–(6.7) and the assumption (AT1) imply
11
15
29
3+ε
σ̄n2 log
n
n
n−1−δ
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
30
1
D. BELOMESTNY
for some δ > 0 and ξ > ξ0 . Noting that
σ̄n2 ≤ C
3
−1/ hn
4
we derive
6
7
8
9
10
17
18
19
20
n
(6.9)
1
2π
R
|1 − K(uhn )||ψk (u)| du hβn
hβn
28
29
30
31
Introduce
2π
43
12
13
{|u|>aK / hn }
R
K(u) =
(6.10)
15
16
17
18
20
21
(1 + |u| )|F[ν̄k ](u)| du,
n → ∞.
β
22
23
24
25
26
27
28
29
30
31
33
34
−1
35
R
36
37
e−iuz K(z) dz.
38
The assumption (AK) together with the smoothness of K implies that K(z) has
finite absolute moments up to order m ≥ s and it holds
(6.11)
14
19
|u|β |F[ν̄k ](u)| du
then by the Fourier inversion formula
37
42
35
41
11
32
34
40
10
1 1 iuz
e K(u) du,
K(z) =
33
39
9
1
νk (x0 ) − ν̄k (x0 ) =
e−iux0 ψk (u)K(uhn ) du − ν̄k (x0 )
2π R
1
k,2 − ψ (u) K(uhn ) du
+
e−iux0 ψ
k
2π R
= J1 + J2
27
38
8
−1/ hn
6.2. Proof of Theorem 4.7. We have
26
36
7
1
+
|1 − K(uhn )||ψk (u)| du.
2π R
The second, bias term on the r.h.s. of (6.9) can be easily bounded if we recall that
νk ∈ Sβ and K(u) = 1 on [−aK , aK ]
23
32
6
22
25
5
−1/ hn
n
21
24
4
log3+ε n 1/ hn
νk − ν̄k L∞ (R,w) = Oa.s.
R2k (u) du
13
16
3
Other terms in R0 can be studied in a similar manner. Finally,
12
15
2
R2k (u) du,
log3+ε n 1/ hn
2
Rk (u) du .
sup[w(|z|)|S (z)|] = Oa.s.
z∈R
11
14
1
1/ hn
2
5
F:aos901.tex; (Laima) p. 30
K(z) dz = 1,
z K(z) dz = 0,
k
k = 1, . . . , m.
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 31
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
Hence
2
J1 =
3
4
and
6
+ 8
9
with CK =
17
18
19
21
22
R
27
36
37
38
39
40
41
42
43
|K(v)| dv ≤ Cν̄ CK (hn /δ)
13
14
Further, by Taylor expansion formula,
K(v)
16
17
18
x +hn v (s)
0
ν̄k (ζ )(ζ − x0 )s−1
(s − 1)!
x0
15
dζ dv 19
20
21
22
23
24
25
26
27
28
I21 ≤
31
35
|v|>δ/ hn
Hence,
29
34
9
12
m
s−1 j (j )
h ν̄ (x ) n k
0
j
K(v)v dv .
I21 = j!
|v|>δ/ hn
j =0
26
8
11
First, let us bound I21 from above. Note that, due to (6.11),
25
33
|v|≤δ/ hn
7
10
= I21 + I22 .
23
32
6
[ν̄k (x0 ) − ν̄k (x0 + hn v)]K(v) dv |K(v)||v|m dv.
+ 20
30
5
s−1 j (j )
h ν̄ (x ) n k
0
j
I2 ≤ K(v)v dv j
!
|v|≤δ/
h
n
j =0
16
28
|v|≤δ/ hn
I1 ≤ 2Cν̄
14
24
4
[ν̄k (x0 ) − ν̄k (x0 + hn v)]K(v) dv Since ν̄∞ ≤ Cν̄ for some constant Cν̄ > 0, we get
13
15
3
= I1 + I2 .
10
12
−∞
2
ν̄k (x0 + hn v)K(v) dv − ν̄k (x0 )
|v|>δ/ hn
7
11
1
∞
|J1 | ≤ 5
31
j (j )
δ |ν̄k (x0 )|
hn m s−1
δ
m
j =0
j!
|v|>δ/ hn
29
|K(v)||v|m dv
hn
LCK exp(δ).
δ
Furthermore, we have for I22
Lhsn
I22 ≤
|K(v)||v|s dv.
s! |v|≤δ/ hn
Combining all previous inequalities and taking into account the fact that m ≥ s,
we derive
≤
|J1 | hsn ,
n → ∞.
The stochastic term J2 can handled along the same lines as in the proof of Theorem 4.4.
30
31
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
32
1
D. BELOMESTNY
6.3. Proof of Theorem 4.9. Define
2
K0 (x) =
3
4
5
6
7
8
9
10
11
12
2
3
ak x
4
is well defined on R. Next, fix two positive numbers β and γ such that γ ∈ (0, 1)
and 0 < β < 1 − γ . Consider a function
(u) =
15
19
−∞
μ(x + zh)K(z) dz
27
28
29
30
31
32
33
34
35
36
−∞
39
40
41
42
43
11
12
15
17
18
20
21
22
23
In the next lemma, some properties of the functions μ and μh are collected.
24
25
L EMMA 6.1.
Functions μ and μh have the following properties:
(6.12)
−n
max{μ(x), μh (x)} |x|
,
28
29
|x| → ∞,
that is, both functions μ(x) and μh (x) decay faster than any negative power of x,
(iii) it holds
(6.13)
x02 μ(x0 ) − x02 μh (x0 ) ≥ Dhβ
26
27
(i) μ and μh are uniformly bounded on R,
(ii) for any natural n > 0
log
−1
(1/ h)
for some constant D > 0 and h small enough.
37
38
10
19
25
26
9
16
∞
2π
23
8
14
1 ∞ −ixu
e
(u) du.
μ(x) =
22
7
(1 + u2 )(1+β)/2 log2 (e + u2 )
for any h > 0, where
21
6
13
μh (x) =
18
5
eix0 u
for some x0 > 0 and define
17
24
∞
&
sin(ak x) 2
with ak = 2−k , k ∈ N. Since K0 (x) is continuous at 0 and does not vanish there,
the function
1 sin(2x) K0 (x)
K(x) =
2π πx K0 (0)
14
20
1
k=1
13
16
F:aos901.tex; (Laima) p. 32
30
31
32
33
34
35
36
37
Fix some ε > 0 and consider two functions
1−ε
ν1 (x) = νγ (x) +
+ εμ(x),
(1 + x 2 )2
1−ε
ν2 (x) = νγ (x) +
+ εμh (x),
(1 + x 2 )2
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 33
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
4
5
6
7
8
9
10
11
12
where νγ (x) is given by
15
16
17
1
1
1
1
1{x ≥ 0} + 1+γ 1{x < 0} .
2
1+γ
(1 + x ) x
|x|
Due to the statements (i) and (ii) of Lemma 6.1, one can always choose ε in such
a way that ν1 and ν2 stay positive on R+ and thus they can be viewed as the Lévy
densities of some Lévy processes L1,t and L2,t , respectively. It directly follows
from the definition of ν1 and ν2 that ν1 , ν2 ∈ Bγ . The next lemma describes some
other properties of ν1 (x) and ν2 (x). Denote ν̄1 (x) = x 2 ν1 (x) and ν̄2 (x) = x 2 ν2 (x).
νγ (x) =
L EMMA 6.2.
20
21
−∞
(1 + |u|β )|F[ν̄i ](u)| du < ∞,
i = 1, 2,
that is, both functions ν1 (x) and ν2 (x) belong to the class Sβ .
Let us now perform a time change in the processes L1,t and L2,t . To this end, introduce a time change T (t), such that the Laplace transform of T (t) has following
representation
28
29
30
31
χ (P , Q) =
2
34
35
38
39
40
41
42
43
0
9
15
16
17
L EMMA 6.3.
fulfills
(6.16)
⎧ dP
⎨
⎩
dQ
+∞,
2
−1
19
20
21
23
y ∈ R+ ,
24
25
26
1,t and p
2,t the marginal densities of the resulting timefor any t ≤ s. Denote by p
changed Lévy processes Y1,t = L1,T (t) and Y2,t = L2,T (t) , respectively. The following lemma provides us with an upper bound for the χ 2 -divergence between
1,t and p
2,t , where for any two probability measures P and Q the χ 2 -divergence
p
between P and Q is defined as
33
37
8
22
e−zy dFt (y),
1 − Ft (y) ≤ 1 − Fs (y),
32
36
∞
where (Ft , t ≥ 0) is a family of distribution functions on R+ satisfying
26
27
Lt (z) = E e−zT (t) =
23
25
7
18
22
24
6
13
18
19
5
14
∞
(6.15)
4
12
x∈R
and
3
11
sup |ν̄1 (x) − ν̄2 (x)| ≥ εDhβ log−1 (1/ h)
(6.14)
2
10
Functions ν̄1 (x) and ν̄2 (x) satisfy
13
14
33
27
28
29
30
31
32
dQ,
if P Q,
otherwise.
Suppose that the Laplace transform of the time change T (t)
33
34
35
36
37
(k+1)
(k)
L
(z)/L (z) = O(1),
|z| → ∞,
for k = 0, 1, 2, and uniformly in ∈ [0, 1]. Then
1, , p
2, ) −1 [L (ch−γ )]2 h(2β+1) ,
χ 2 (p
with some constant c > 0.
38
39
40
h → 0,
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
34
1
2
3
4
D. BELOMESTNY
The proofs of Lemmas 6.1, 6.2 and 6.3 can be found in the preprint version of
our paper Belomestny (2010a). Combining Lemma 6.3 with the inequality (6.14)
and using the well known Assouad’s lemma [see, e.g., Theorem 2.6 in Tsybakov
(2004)], one obtains
'
5
lim inf inf
6
sup
n→∞ ν ν∈B ∩S
γ
β
7
8
F:aos901.tex; (Laima) p. 34
(
P sup |ν̄(x) − ν (x)| > hβn log−1 (1/ hn ) > 0
x∈R
n → ∞.
11
12
17
18
19
20
21
22
25
26
27
32
33
34
(7.1)
αZ (j ) ≤ αT (j ),
j ∈ N.
37
E[φ(Z1 , . . . , Zk )ψ(Zk+l , . . . , Zn )]
40
41
42
43
18
19
20
21
24
25
26
27
28
k, l ∈ N,
1, . . . ,
for any two functions φ : Rk → [0, 1] and ψ : Rn−l−k → [0, 1], where φ(t
tk ) = E[φ(Lt1 , . . . , Ltk )] and ψ(t1 , . . . , tk ) = E[ψ(Lt1 , . . . , Ltk )]. This implies that
the sequence Zk is strictly stationary and α-mixing with the mixing coefficients
satisfying (7.1). 29
30
31
32
33
34
35
7.2. Exponential inequalities for dependent sequences. The following theorem can be found in Merlevéde, Peligrad and Rio (2009).
38
39
17
23
P ROOF. Fix some natural k, l with k + l < n. Using the independence of increments of the Lévy process Lt and the fact that T is a nondecreasing process,
1 , . . . , Tk )] and
we get E[φ(Z1 , . . . , Zk )] = E[φ(T
1 , . . . , Tk )ψ(T
k+l , . . . , Tn )],
= E[φ(T
16
22
35
36
10
15
29
31
9
14
L EMMA 7.1. Let Lt be a d-dimensional Lévy process with the Lévy measure
ν and let T (t) be a time change independent of Lt . Fix some > 0 and consider
two sequences Tk = T (k) − T ((k − 1)) and Zk = Yk − Y(k−1) , k = 1, . . . , n,
where Yt = LT (t) . If the sequence (Tk )k∈N is strictly stationary and α-mixing with
the mixing coefficients (αT (j ))j ∈N , then the sequence (Zk )k∈N is also strictly stationary and α-mixing with the mixing coefficients (αZ (j ))j ∈N , satisfying
28
30
6
13
7.1. Some results on time-changed Lévy processes.
23
24
5
12
15
16
4
11
7. Auxiliary results.
13
14
3
8
2 (2β+1)
n−1 [Lt (c · h−γ
= O(1),
n )] hn
10
2
7
for any sequence hn satisfying
9
1
36
37
38
T HEOREM 7.2. Let (Zk , k ≥ 1) be a strongly mixing sequence of centered
real-valued random variables on the probability space (, F , P ) with the mixing
coefficients satisfying
(7.2)
α(n) ≤ ᾱ exp(−cn),
n ≥ 1, ᾱ > 0, c > 0.
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 35
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
4
5
6
7
8
9
35
Assume that supk≥1 |Zk | ≤ M a.s., then there is a positive constant C depending
on c and ᾱ such that
$ n
%
Cζ 2
Zi ≥ ζ ≤ exp −
P
nv 2 + M 2 + Mζ log2 (n)
i=1
12
13
14
v = sup E[Zi ] + 2
2
2
i
Cov(Zi , Zj ) .
j ≥i
ρj = E Zj2 log2(1+ε) (|Zj |2 ) ,
25
26
27
28
29
j
for some constant C > 0, provided (7.2) holds. Consequently, the following inequality holds
v 2 ≤ sup E[Zi ]2 + C max ρj .
The proof can be found in Belomestny (2010a).
7.3. Bounds on large deviations probabilities for weighted sup norms. Let
Zj = (Xj , Yj ), j = 1, . . . , n, be a sequence of two-dimensional random vectors
and let Gn (u, z), n = 1, 2, . . . , be a sequence of complex-valued functions defined
on R2 . Define
n
1
1 (u) =
m
Xj Gn (u, Xj ),
n j =1
30
31
32
n
1
2 (u) =
m
Yj Gn (u, Xj ),
n j =1
33
34
35
36
3 (u) =
m
37
38
39
4 (u) =
m
40
41
42
43
j
i
22
24
5
7
8
9
P ROPOSITION 7.4.
12
13
14
15
Cov(Zi , Zj ) ≤ C max ρj
j ≥i
21
23
j = 1, 2, . . . ,
with arbitrary small ε > 0 and suppose that all ρj are finite. Then
17
20
4
11
Denote
16
19
3
10
C OROLLARY 7.3.
15
18
2
6
for all ζ > 0 and n ≥ 4, where
10
11
1
n
1
X 2 Gn (u, Xj ),
n j =1 j
n
1
Xj Yj Gn (u, Xj ).
n j =1
Suppose that the following assumptions hold:
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
36
1
2
D. BELOMESTNY
(AZ1) The sequence Zj , j = 1, . . . , n, is strictly stationary and is α-mixing
with mixing coefficients (αZ (k))k∈N satisfying
3
αZ (k) ≤ ᾱ0 exp(−ᾱ1 k),
4
5
6
7
8
9
14
4
31
μ̄n /σ̄n2
18
= O(1),
μ̄n /σ̄n = O(n
Ln /σ̄n = O(n3/2 ),
E[|X|4 |Gn (u, X)|2 ],
1/2−δ/2
σ̄n2
),
42
43
24
25
26
Let w be a symmetric, Lipschitz continuous, positive, monotone decreasing on R+
function such that
30
0 < w(z) ≤ log−1/2 (e + |z|),
$
36
41
23
28
29
(7.4)
P log−(1+ε) (1 + μ̄n )
31
32
z ∈ R.
Then there is δ > 0 and ξ0 > 0, such that the inequality
40
22
27
34
39
= O(n),
n → ∞,
(7.3)
38
20
21
33
37
19
for some δ satisfying 2/p < δ ≤ 1.
32
35
14
16
are uniformly bounded on R by σ̄n2 . Moreover, assume that the sequences μ̄n , Ln
and σ̄n fulfill
27
30
(u, z) ∈ R2 ,
E[|X|2 |Y |2 |Gn (u, X)|2 ]
26
29
13
17
E[(|X|2 + |Y |2 )|Gn (u, X)|2 ],
25
28
9
11
and all the functions
21
24
8
15
|Gn (u, z)| ≤ μ̄n ,
20
23
7
12
19
22
6
where a, b are two nonnegative real numbers not depending on n and the sequence
Ln does not depend on u.
(AG2) There are two sequences μ̄n and σ̄n , such that
16
18
5
10
15
17
2
|Gn (u1 , z) − Gn (u2 , z)| ≤ Ln (a + b|z|)|u1 − u2 |,
11
13
1
3
k ∈ N,
for some ᾱ0 > 0 and ᾱ1 > 0.
(AZ2) The r.v. Xj and Yj possess finite absolute moments of order p > 2.
(AG1) Each function Gn (u, z), n ∈ N is Lipschitz in u with linearly growing
(in z) Lipschitz constant, that is, for any u1 , u2 ∈ R
10
12
F:aos901.tex; (Laima) p. 36
33
34
%
n
k − E[m
k ]L∞ (R,w) > ξ ≤ Bn−1−δ
m
2
σ̄n log n
holds for any ξ > ξ0 , any k ∈ {1, . . . , 4}, some positive constant B depending on ξ
and arbitrary small ε > 0.
The proof of the proposition can be found in Belomestny (2010a).
35
36
37
38
39
40
41
42
43
AOS imspdf v.2011/08/03 Prn:2011/08/05; 13:43
F:aos901.tex; (Laima) p. 37
STATISTICAL INFERENCE FOR TIME-CHANGED LÉVY PROCESSES
1
2
3
4
5
6
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8
9
10
<uncited>
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<uncited>
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D UISBURG -E SSEN U NIVERSITY
F ORSTHAUSWEG 2, D-47057 D UISBURG
G ERMANY
E- MAIL : [email protected]
<unstr>
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F:aos901.tex; (Laima) p. 39
THE ORIGINAL REFERENCE LIST
The list of entries below corresponds to the original Reference section of your article. The
bibliography section on previous page was retrieved from MathSciNet applying an automated
procedure.
Please check both lists and indicate those entries which lead to mistaken sources in automatically generated Reference list.
O. Barndorff-Nielsen (1998). Processes of normal inverse Gaussian type. Finance Stoch., 2(1), 41–
68.
I. Basawa and P. Brockwell (1982). Nonparametric estimation for nondecreasing Lévy processes, J.
Roy. Statist. Soc. Ser. B, 44, 262–269.
F. Bauer and M. Reiß (2008). Regularisation independent of the noise level: an analysis of quasioptimality. Inverse Problems, 24(5).
D. Belomestny (2010). Statistical inference for multidimensional time-changed Lévy processes based
on low-frequency data, arXiv:1003.0275.
D. Belomestny (2010). Spectral estimation of the fractional order of a Lévy process. Annals of Statistics, 38(1), 317–351.
M. Bismut (1983). Calcul des variations stochastique et processus de sauts. Zeitschrift für
Wahrscheinlichke itstheories Verw. Gebiete, 63, 147–235.
S. Bochner (1949). Diffusion Equation and Stochastic Process. Proc. N.A.S. USA, 35.
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P. Carr, H. Geman, D. Madan, and M. Yor (2003). Stochastic volatility for Lévy processes. Mathematical Finance, 13, 345-382.
P. K. Clark (1973). A subordinated stochastic process model with fixed variance for speculative
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META DATA IN THE PDF FILE
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Following information will be included as pdf file Document Properties:
Title
: Statistical inference for time-changed L\351vy processes
via composite characteristic function estimation
Author : Denis Belomestny
Subject : The Annals of Statistics, 0, Vol.0, No.00, 1-38
Keywords: 62F10, 62J12, 62F25, 62H12, Time-changed Levy processes,
dependence, pointwise and uniform rates of convergence, composite
function estimation
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