Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Functional limit theorems for
multiparameter fractional Brownian motion
For proofs see http://arxiv.org/abs/math.PR/0405009
Anatoliy Malyarenko
Department of Mathematics and Physics
Mälardalen University
SE 721 23 Västerås, Sweden
July 28, 2004
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
1
The formulation of the problem
2
The description of the Strassen’s ball
3
The main theorem
4
Examples
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The multiparameter fractional Brownian motion
(FBM)
Definition
The multiparameter fractional Brownian motion with the Hurst
parameter H ∈ (0, 1) is the centred Gaussian random field ξ (x)
on the space RN with the covariance function
R (x, y) = Eξ (x)ξ (y)
1
= (kxk2H + kyk2H − kx − yk2H ).
2
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The cloud of normed increments
Let t0 be a real number. Let for every t ≥ t0 there exists a
non-empty set of indices J (t ). Let every element j ∈ J (t )
defines the vector yj ∈ RN and the positive real number uj . Let
B = { x ∈ RN : kxk ≤ 1 } and
P(t ) = { (yj , uj ) : j ∈ J (t ) }.
Definition
The cloud of normed increments is
(
S (t ) =
ξ (y + ux) − ξ (y)
p
η(x) =
: (y, u ) ∈ P(t )
2h(t )u H
Anatoliy Malyarenko
)
⊂ C (B).
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The problem
Describe the set of P-a. s. limit points of the cloud of normed
increments S (t ) as t → ∞ under some restrictions on h(t ) and
P(t ).
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The problem
Describe the set of P-a. s. limit points of the cloud of normed
increments S (t ) as t → ∞ under some restrictions on h(t ) and
P(t ).
Hint: In the case of the ordinary Brownian motion (N = 1,
H = 1/2) and when h(t ) = log log t and P(t ) = {(0, t )}, the
answer is given by the functional law of the iterated logarithm
(Strassen, 1964): the limit set is Strassen’s ball
K =
f ∈ C [0, 1] :
1
Z
0
2
(f (t )) dt ≤ 1 .
0
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Notation
Let
λm1 ≥ λm2 ≥ · · · ≥ λmn ≥ · · · > 0
be the sequence of all eigenvalues (with multiplicities) of the
positive definite kernel
Γ(m − H )
π N /2
bm (r , s) =
(r 2H + s2H )δm0 −
(rs)m (r + s)2(H −m
Γ(N /2 + m)
Γ(−H )
4rs
× 2 F1 m + (N − 1)/2, m − H ; 2m + N − 1;
(r + s)2
in the Hilbert space L2 ([0, 1], dr ). Let ψmn (r ) be the eigenbasis
of the kernel bm (r , s).
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The Strassen’s ball for the multiparameter FBM
Lemma
The Strassen’s ball K for the multiparameter FBM is the
centred unit ball in the Hilbert space
(
H =
∞
f ∈ C (B) : kf k2S =
∞ h(m,N )
∑∑ ∑
m=0 n=1 l =1
l
(fmn
)2
<∞
λmn
)
,
l
where fmn
are the Fourier coefficients of f with respect to the
basis
l
{ ψmn (r )Sm
(ϕ, ϑ1 , . . . , ϑN −2 ) : m ≥ 0, n ≥ 1, 1 ≤ l ≤ h(m, N ) }.
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
More notation
Let Rr (yj , uj ) be the cylinder
Rr (yj , uj ) = { (y, u ) : ky − yj k ≤ ruj , e−r uj ≤ u ≤ er uj },
r > 0.
Let Fr (t ) be the volume of the union of all cylinders Rr (yj , uj )
that are defined before the moment t, with respect to the
measure u −N −1 dy du.
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The main theorem
Theorem
Let the function h(t ) : [t0 , ∞) 7→ R satisfies the next conditions:
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The main theorem
Theorem
Let the function h(t ) : [t0 , ∞) 7→ R satisfies the next conditions:
+ h(t ) is increasing and lim h(t ) = ∞.
t →∞
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The main theorem
Theorem
Let the function h(t ) : [t0 , ∞) 7→ R satisfies the next conditions:
+ h(t ) is increasing and lim h(t ) = ∞.
t →∞
Z ∞
+ The integral
e−ah(t ) dF1 (t ) converges for a > 1 and
t0
diverges for a < 1.
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
The main theorem
Theorem
Let the function h(t ) : [t0 , ∞) 7→ R satisfies the next conditions:
+ h(t ) is increasing and lim h(t ) = ∞.
t →∞
Z ∞
+ The integral
e−ah(t ) dF1 (t ) converges for a > 1 and
t0
diverges for a < 1.
Then, in the uniform topology, the set of P-a. s. limit points of
the cloud of increments S (t ) as t → ∞ is Strassen’s ball K .
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Example (Local functional law of the iterated logarithm)
Let t0 = 3. Let J (t ) contains only one element 0. Let y0 = 0
and u0 = t −1 . The function h(u ) = log log u satisfies the
conditions of Theorem. We obtain, that, in the uniform topology,
the set of P-a. s. limit points of the cloud of increments
ξ (tx)
p
2 log log t −1 t H
as t ↓ 0 is Strassen’s ball K . For the case of N = 1 and
H = 1/2, this result is due to (Gantert, 1993).
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Example (Local law of the iterated logarithm for the FBM)
Let F (f ) = kf k∞ , f ∈ C (B). On the one hand, we have
|ξ (t )|
= sup kf k
lim sup p
2 log log t −1 t H
f ∈K
t ↓0
P − a.s.
On the other hand, according to (Benassi et al, 1997), we have
|ξ (t )|
lim sup p
=1
2 log log t −1 t H
t ↓0
P − a.s.
It follows that
sup kf k = 1.
f ∈K
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
(1)
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Example (Global functional law of the iterated logarithm)
Let t0 = 3. Let J (t ) contains only one element 0. Let y0 = 0
and u0 = t. It is easy to check, that dA1 (t ) is comparable to
t −1 dt. It follows that, in the uniform topology, the set of P-a. s.
limit points of the cloud of increments
√
ξ (tx)
2 log log tt H
as t → ∞ is Strassen’s ball K . For the case of N = 1 and
H = 1/2, this result is due to (Strassen, 1964).
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Example (Global law of the iterated logarithm for the FBM)
Using the continuous functional F (f ) = kf k∞ , we obtain
|ξ (x)|
= sup kf k
lim sup √
t →∞ kxk≤t
2 log log tt H
f ∈K
P − a.s.
or, by (1),
|ξ (x)|
lim sup √
=1
2 log log tt H
t →∞ kxk≤t
Anatoliy Malyarenko
P − a.s.
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Example (Functional Lévy modulus of continuity)
Let t0 = 2. Let J (t ) = { y ∈ RN : kyk ≤ 1 − t −1 } and u = t −1
for any y ∈ J (t ). The function h(u ) = N log u satisfies the
conditions of Theorem. It follows that, in the uniform topology,
the set of P-a. s. limit points of the cloud of increments
(
S (t ) =
ξ (y + tx) − ξ (y)
η(x) = p
: kyk ≤ 1 − t
)
2N log t −1 t H
as t ↓ 0 is Strassen’s ball K . For the case of N = 1 and
H = 1/2, this result is due to (Mueller, 1981).
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Application to approximation theory
Let µ be the Gaussian measure on the space C (B) that
corresponds to the multiparameter FBM. Let L be the linear
space of all deterministic functions f ∈ C (B) satisfying the
condition
|f (x + y) − f (x)|
< ∞.
lim sup sup p
2N log kyk−1 kykH
kyk↓0 x∈B
It follows from functional Lévy modulus of continuity that
µ(L) = 1. By (Lifshits, 1995) H ⊂ L. By Lemma, we obtain the
following Bernstein–type theorem from approximation theory:
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM
Outline
The formulation of the problem
The description of the Strassen’s ball
The main theorem
Examples
Bernstein–type theorem
Theorem
Let f ∈ C (B) with f (0) = 0 has the following constructive
property:
∞
∞ h(m,N )
∑∑ ∑
m=0 n=1 l =1
l
(fmn
)2
< ∞.
λmn
Then f has the following descriptive property:
|f (x + y) − f (x)|
lim sup sup p
< ∞.
2N log kyk−1 kykH
kyk↓0 x∈B
Anatoliy Malyarenko
Functional limit theorems for multiparameter FBM