UDC 519.213 519.214
On Multivariate Geometric Random Sums
I.V. Zolotukhin, Russian Academy of Sciences, P.P. Shirshov Institute of
Oceanology, St. Petersburg Department
[email protected]
Abstract
In this paper, we use multivariate geometric distribution to generalize
the notion of geometric random sum to the multidimensional case.
The characteristic functions of multivariate random sums are found as
well as their projections on an arbitrary coordinate hyperplane. The sufficient conditions for weak convergence of these sums to the Marshall-Olkin
multivariate exponential distribution and to the multivariate generalized
Laplace distribution are given.
Keywords: multivariate geometric distribution, Marshall-Olkin multivariate exponential distribution, multivariate generalized Laplace distribution, characteristic function.
Introduction
Geometric summation arises naturally in the number of fields such as economics,
physics, biology, queuing theories. Properties of geometric random sums of the
form
M
X
Xj ,
(1)
j=1
where Xj are i.i.d. random variables, M is a random variable with the
geometric distribution: P (M = n) = pq n−1 , q = 1 − p, (n = 1, 2, . . . ); M and
Xj (j = 1, 2, . . . ) are independent, have been well studied [1].
It was found that the asymptotic behavior of these sums depends on the
expectation a = EXj . If Xj ≥ 0 and 0 < a < ∞, then the limit distribution is
exponential, and if a = 0 and 0 < EXj2 < ∞, then the limit distribution is the
Laplace distribution (see [2], p.86), both after a suitable normalization.
In practice, often occur time series of random vectors. To date have been
studied those limit distributions that are approximated by geometric sums in
the form (1) for i.i.d. random vectors (see, for example, [3] and [4]). However, in
the time series of random vectors occurring, for example, in models of financial
mathematics, the number of random variables can be different for different components, and these random numbers are dependent on each other. For example,
if we consider the time series the value of portfolios investments, or other assets,
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then each component of the daily changes of these values is the sum of a random
number of random changes, some of which affect only one of the components,
while others affect to several ones. The result could be a vector sum in the form


Mk
M1
X
X
(j)
(j)
Z = (Z1 , . . . , Zk ) = 
X1 , . . . ,
Xk  ,
j=1
j=1
wherein the random indices Ml (l = 1, . . . , k) dependent with each other. We
consider this case. We introduce a new class of multivariate geometric random
sums.
Let E = {ε} is the set of k - dimensional indices ε = (ε(1) , . . . , ε(k) ), each
coordinate of which is equal to 0 or 1, and El is the set of indices for which
ε(l) = 1.
Let Nε are the independent geometrically distributed random variables:
P (Nε = n) = pε qεn−1 , n = 1, 2, . . . , pε = 1 − qε ,
Ml = min {Nε } , l = 1, . . . , k.
(2)
ε ∈ El
The distribution of the vector M = (M1 , . . . , Mk ) is introduced and studied
in [5], and is called multivariate geometric distribution (MVG).
Survival function of vector M is
Y
max ε m
P̄ (m) = P̄ (m1 , ... , mk ) = P min {Nε } >m1 , ... , min {Nε } >mk =
.
qε i
ε ∈ Ek
ε ∈ E1
ε∈E
Here m = (m1 , . . . , mk ), mj (j = 1, . . . , k) are integers, εm is the coordinatewise product of vectors ε and m (here and below, a couple of signs like αβ,
standing side by side, will mean their coordinate-wise product).
Multivariate geometric distribution has properties similar to those of onedimensional geometrical laws. In particular, it has property of absence of aftereffect at the shift of all coordinates on the same value n:
P (M > n + m/M >m ) = P (M > m), n = (n, . . . , n).
Generalized multivariate geometric random sum is called a random
vector sum of the form


Mk
M1
X
X
(j)
(j)
Z = (Z1 , . . . , Zk ) = 
X1 , . . . ,
Xk  ,
(3)
j=1
j=1
(j)
where Ml are defined above in (2), Xr (r = 1, . . . , k) are independent
random variables identically distributed for each l with the known characteristic
function
E exp(i tr Xr ) = φr (tr ),
2
(j)
and Ml and Xr are independent.
Note that the dependance between the components of the vector Z in the
formula (3) is due to dependence in the summation indices Ml , and not because
(j)
of the dependence between the coordinates of the vectors Xl .
Multivariate geometric random sums include the two extreme cases.
For Ml = N1 (l = 1, . . . , k), Nε = 0, ε 6= 1, we have the standard geometric
vector sums.
And for Ml = Nεl , where εl = (0, . . . , 0, 1, 0, . . . , 0), Nε = 0 for ε 6= εl ,
l
each component will be a univariate geometric random sum in the form (1), and
components of vector Z are independent.
Some details about Multivariate Exponential Distribution and Multivariate Laplace Distribution
The random vector V = (V1 , . . . , Vk ) having Marshall-Olkin multivariate exponential distribution (see [6]) is given by the survival function
!
X
F̄ (z) = P (V1 > z1 , . . . , Vk > zk ) = exp −
λε max εz , zi > 0,
ε∈E
1≤i≤k
here (λε ≥ 0, ε ∈ E) is a compact notation for the distribution parameters. The
class of these distributions will be denoted as M V E(λε , ε ∈ E).
Explicit expressions for the characteristic function of the random vector V ∈
M V E(λε , ε ∈ E) as well as its projections to arbitrary coordinate hyperplane,
have been received in [8]. If to use notations of the present paper, they will be
written down as:
X
1
λε ΨV (εt),
ΨV (t) = P
λε − i(t, 1)
ε∈E
ε∈E
ΨV (εt) =
P
X
1
λδ ΨV (δεt).
λδ − i(t, ε)
δ:δε>0
δ:δε>0
The multivariate generalized Laplace distribution was introduced in [7].
This is a mixture by parameter scale for the k-dimensional normal vector with
independent components and zero expectations, if mixing distribution is the
Marshall-Olkin multivariate exponential distribution. Random vector with this
distribution can be defined as follows:
Let Y = (Y1 , Y2 , . . . , Yk ) is a normal vector with zero expectation and identity covariance matrix, and V = (V1 , . . . , Vk ), V√∈ MVE (λ√
ε , ε ∈ E).
Then the distribution of the vector W = ( V1 Y1 , . . . , Vk Yk ) shell be the
multivariate generalized Laplace distribution denoted by M GLD(λε , ε ∈
E).
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The characteristic function of the vector W is
X
1
ΨW (t) = P
λε ΨW (εt).
1
λε + 2 (t, t) ε∈E
ε∈E
Let denote
P
Φ(t) = P
ε∈E
λε
ε∈E
λε + 12 (t, t)
.
It is the characteristic function of the multivariate symmetric Laplace distribution.
Now we can write down ΨW (t) as
X
ΨW (t) = Φ(t) pε ΨW (εt),
ε∈E
where
λε
pε = P .
λε
ε∈E
Hence the distribution of W is the discrete mixture of multivariate symmetrical Laplace distribution with their convolutions with projections of W on the
coordinate hyperplanes.
When λ1 > 0 and λε = 0, ε 6= 1, ε ∈ E, we have the symmetric Laplace
distribution.
When λ0,...,0,1,0,...,0 > 0 (l = 1, . . . , k) and λε = 0 for other cases (ε ∈ E),
l
the coordinates of W will be independent random variables with the univariate
Laplace distribution.
The characteristic functions of projection of vector W on the coordinate
hyperplane ε are
ΨW (εt) =
P
δ:δε>0
X
1
λδ ΨW (δεt).
1
λδ + 2 (εt, εt) δ:δε>0
Main Results
Let denote ΨZ (t) = Eexp(i(t, Z)) the characteristic function of vector Z; ln φ(t) =
(ln φ1 (t1 ), . . . , ln φk (tk )) is the coordinate-wise logarithm of the vector function
φ(t) = (φ1 (t1 ), . . . , φk (tk )); t = (t1 , . . . , tk ); ε1 ∨ ε2 is the vector, each i-th
(i) (i)
coordinate is max{ε1 , ε2 }.
Define the partial order on the set E by the rule:
∀ε, δ ∈ E δ 6 ε, if ∀j (j = 1, . . . , k) δj 6 εj .
Let δ < ε if δ 6 ε and δ 6= ε. Note that ∀δ, ε ∈ E
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δε 6 ε.
Theorem 1
The characteristic function of the projection of vector Z on the coordinate
hyperplane ε is
ΨZ (εt) = Eexp(i(t, εZ)) =
j
k
X
X Y
exp(ε, ln ϕ(t))
·
=
pδl
Q
1 − exp(ε, ln ϕ(t))
qγ j=1 δl :δl ε>0 l=1
γ:γε>0
l=1,...,j

Y
qγ ΨZ 
γ:γε>0
γ6=δl
l=1,...,j
j
_

δl εt .
l=1
Corollary 1.1
ΨZ (t) = Eexp(i(t, Z)) =
j
k
X
X Y
exp(1, ln ϕ(t))
Q
=
pδl
·
1 − exp(1, ln ϕ(t)) δ∈E qγ j=1
δl ∈E l=1
l=1,...,j

Y
qγ ΨZ 
γ∈E
γ6=δl
l=1,...,j
j
_

δl t .
l=1
Corollary 1.2
The characteristic function of the vector Z projection on the axis ε = (1, 0, . . . , 0)
is
Q
ϕ1 (t1 ) 1 − δ∈E1 qδ
Q
Ψ(εt) = Eexp(i(t1 , Z1 )) =
.
1 − ϕ1 (t1 ) δ∈E1 qδ
Now let denote pε = λε p, ε ∈ E. We shall search the limit distributions of
the vector Z as p → 0.
Theorem 2
(j)
Let Xl
(j)
≥ 0; EXl
p
p
Z1 , . . . , Zk
a1
ak
= al ; 0 < al < ∞; l = 1, . . . , k, then


Mk
M1
X
X
p (j)  D
p (j)
X1 , . . . ,
Xk
−−−→ V, V ∈ M V E(λε , ε ∈ E)
=
p→0
a
a
1
j=1
j=1 k
Theorem 3
(j)
Let EXl
= 0; EXl2 = σl2 ; 0 < σl2 < ∞; l = 1, . . . , k, then
√
p
σ1
√
Z1 , . . . ,
p
σk
Zk
D
−−−→ W, W ∈ M GLD(λε , ε ∈ E).
p→0
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Conclusions
We present the following results:
- The characteristic function of the vector Z as well as projections of Z on
any coordinate hyperplane is found.
- It has been shown that the distributions of the vector Z can be recursively
restored by distributions of their univariate components.
- It has been shown that, by analogy with the univariate case, after a suitable normalization the limit distributions will be Marshall-Olkin distribution,
(j)
(j)
if Xl ≥ 0 and 0 < EXl < ∞, and multivariate Laplace distribution, if
(j)
(j) 2
EXl = 0 and 0 < E(Xl ) < ∞.
- Thus, it is shown that these distributions possess the property of geometric
stability, under the special scheme of geometric summation.
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