Functional iterated logarithm law for a Wiener process
The functional iterated logarithm law for a Wiener process in the Bulinskii form for great and small times is proved.
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| Cite this: | Functional iterated logarithm law for a Wiener process / D.S. Budkov, S.Ya. Makhno // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 22–28. — Бібліогр.:3 назв.— англ. |
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| author | Budkov, D.S. Makhno, S.Ya. |
| author_facet | Budkov, D.S. Makhno, S.Ya. |
| citation_txt | Functional iterated logarithm law for a Wiener process / D.S. Budkov, S.Ya. Makhno // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 22–28. — Бібліогр.:3 назв.— англ. |
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| description | The functional iterated logarithm law for a Wiener process in the Bulinskii form for great and small times is proved.
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Theory of Stochastic Processes
Vol. 13 (29), no. 3, 2007, pp. 22–28
UDC 519.21
DMITRII S. BUDKOV AND SERGEY YA. MAKHNO
FUNCTIONAL ITERATED LOGARITHM LAW
FOR A WIENER PROCESS
The functional iterated logarithm law for a Wiener process in the Bulinskii form for
great and small times is proved.
1. Introduction. Let w(t) be a d-dimensional Wiener process on the probability
space (Ω,F , P ), t ≥ 0. Introduce the sequence of random processes
ξn(t) =
w(nt)√
nϕ(n)
, n = 3, 4, ... (1)
where ϕ(n) is an arbitrary sequence.
In [1], A.V. Bulinskii proved a version of the Strassen iterated logarithm law in which
the normalizing function ϕ(n) is an arbitrary monotone increasing function. Let us
formulate this result.
Denote, by C([0, 1]; Ed), the space of all continuous functions x(t) on the interval [0, 1]
with values in the Euclidean space (Ed, | · |) and with norm
||x|| = sup
t∈[0,1]
|x(t)|, (2)
Let K1([0, 1]; Ed) be the space of functions x(t) ∈ C([0, 1]; Ed) such that x(0) = 0 and
x(t) =
∫ t
0 ẋ(s)ds for some function ẋ(·) ∈ L2([0, 1]; Ed). We set
||x||K1 = ||ẋ||L2([0,1];Ed). (3)
Following [1], we define Φ as a class of all increasing functions ϕ(t), t ≥ 0, such that
limt→∞ ϕ(t) = ∞. Now we introduce a functional
J(ϕ, r, c) =
∞∑
k=1
exp
{−rϕ2([ck])
2
}
, c > 1, (4)
where [·] denotes the integer part of a number. For every ϕ ∈ Φ, we denote
R2(ϕ) = inf{r > 0 : J(ϕ, r, c) < ∞}, (5)
and R(ϕ) = ∞ if there exists no r < ∞ such that J(ϕ, r, c) < ∞. Let us remark that if
J(ϕ, r, c0) < ∞ for a certain c0 > 1, then J(ϕ, r, c) < ∞ for every c > 1.
Let KR = {x(t) ∈ K1, ||x||K1 ≤ R2}.
In [1, Theorem 1], the following result was proved.
Bulinskii theorem. For ϕ ∈ Φ, the limit set of sequence (1), t ∈ [0, 1], coincides
with KR, and R = R(ϕ) is defined in (5).
2000 AMS Mathematics Subject Classification. Primary 60H10, 60H20.
Key words and phrases. Wiener process, Strassen iterated logarithm law, Bulinskii iterated logarithm
law.
22
FUNCTIONAL ILL FOR A WIENER PROCESS 23
In [1, Theorem 3], it was proved that R can be defined as R = 1
Q , where
Q = lim
t→∞
ϕ(t)√
2 ln ln t
.
This theorem yields the usual functional Strassen iterated logarithm law and allows us to
study the asymptotic behavior of the process ξn(t) for normalizing functions ϕ(n) such
that limt→∞
ϕ(t)√
2 ln ln t
= ∞.
In the present article, we generalize this result in two directions. First, we consider
sequence (1) on the whole time axis and, second, we prove a local version of the Bulinskii
theorem.
2. Functional iterated logarithm law on the whole time axis.
Let C([0,∞); Ed) denote the space of all continuous functions on [0,∞) with values
in Ed. We set
Θ =
{
θ ∈ C([0,∞) : Ed) : θ(0) = 0, lim
t→∞
|θ(t)|
t
= 0
}
.
On this space, we define the metric
||θ||Θ = sup
t≥0
| θ(t)|
1 + t
. (6)
Then (Θ, ||·||Θ) is a separable Banach space. Let H1([0,∞); Ed) be the space of functions
θ ∈ Θ such that θ(t) =
∫ t
0 θ̇(s)ds for some θ̇ ∈ L2([0,∞); Ed). We define
||θ||H1 = ||θ̇||L2([0,∞);Ed). (7)
Lemma. There exists the bijective correspondence between the spaces K1([0, 1]; Ed)
and H1([0,∞); Ed). If f ∈ K1([0, 1]; Rd), then
g(t) =
∫ t
0
ḟ
( s
s + 1
) 1
1 + s
ds ∈ H1([0,∞); Ed). (8)
Conversely, if g ∈ H1([0,∞); Rd), then
f(t) =
∫ t
0
1
1 − s
ġ
( s
1 − s
)
ds ∈ K1([0, 1]; Ed). (9)
Furthermore, ||g||H1([0.∞);Ed) = ||f ||K1([0,1];Ed).
Proof. Let f ∈ K1([0, 1]; Ed). It is necessary to prove that, for the function g(x) from
(8),
lim
t→∞
|g(t)|
t
= 0. (10)
According to the Cauchy–Buniakowski inequality,
|g(t)|2 =
∣∣∣ ∫ t
t+1
0
ḟ(s)
1
1 − s
ds
∣∣∣2 ≤ ||f ||2K1
∫ t
t+1
0
1
(1 − s)2
ds = t||f ||2K1 .
This yields (10). It is easy to verify the equality of norms. Lemma is proved.
In what follows, we will use the Schilder theorem [2, Theorem 1.3.27] on large devia-
tions of a Wiener process with small variance. For convenience, we present its formula-
tion. Define
I(ψ) =
⎧⎨
⎩
0, if ψ /∈ H1([0,∞); Ed),
1
2
||ψ||H1 , if ψ ∈ H1([0,∞); Ed).
24 DMITRII S. BUDKOV AND SERGEY YA. MAKHNO
Schilder theorem. Let Wε(·) be the measure corresponding to a process εw(t) on
the space Θ with a Borel σ-algebra B. Then
a) for any closed set A ∈ B,
lim
ε→0
ε2Wε(A) ≤ − inf{I(ψ); ψ ∈ A}; (11)
b)for any open set B ∈ B,
lim
ε→0
ε2Wε(B) ≥ − inf{I(ψ); ψ ∈ B}. (12)
The functional I(ψ) is called the action functional for a family Wε(·). For any a < ∞,
the set {ψ : I(ψ) ≤ a} is closed in the space (Θ,B).
Introduce a class of functions LR :
LR = {θ(t) ∈ H1 : ||θ||H1([0,∞);Ed) ≤ R2}, (13)
where R = R(ϕ) is defined in (5).
Theorem 1. For ϕ ∈ Φ with probability 1, the set of limit points of sequence (1) is
LR, where R is defined in (5).
Proof. We prove the theorem in three standard steps. Let us denote nk = [ck],
zk(t) = ξnk
(t).
Step 1). We need to prove that, for every R2 < ∞, every c > 1, and every δ > 0,
there exists a positive integer k0 such that, for every k > k0,
ρ(zk,LR) < δ. (14)
We set Nδ = {ψ : ρ(ψ,LR) ≥ δ}. Then there exists η > 0 such that
inf
ψ∈Nδ
I(ψ) ≥ R2
2
+ η.
As the set Nδ is closed, relation (11) yields
P
{
zk ∈ Nδ
}
≤ exp
{
− ϕ2(ck)
(R2
2
+ η
)}
By the definition of the number R, we have
∑
k P
{
zk ∈ Nδ
}
< ∞. By applying the
Borel–Cantelli lemma, we get (14). Step 1) is proved.
Step 2). For R2(ϕ) < ∞, we need to prove that every limit point of the sequence ξn(t)
is an element of LR. If n = nk, thos follows from step 1). Let now n ∈ [nk, nk+1]. Denote
ψ(n) =
√
nϕ(n). Since the function ψ(n) is nondecreasing, we can write
1
ψ(n)
=
αnk
ψ(nk)
+
βnk
ψ(nk+1)
, (15)
where αnk ≥ 0, βnk ≥ 0 and αnk +βnk = 1. Set z̃nk(t) = αnkzk(t)+βnkzk+1(t). We note
that, for large k, the functions z̃nk ∈ {f : ρ(f,LR) < δ}. This follows from the fact that
if the functions x(t), y(t) ∈ LR and α, β ≥ 0, α + β = 1, then αx(t) + βy(t) ∈ LR. The
assertion of step 2) will be proved if we prove the following estimate: for every δ > 0,
there exist a number c > 1 and a positive integer k0 such that, for every k > k0,
sup
t≥0
1
1 + t
sup
n∈[nk,nk+1]
|ξn(t) − z̃nk(t)| < δ (16)
with probability 1. From (15), we have
ξn(t) =
w(nk
n
nk
t)
ψ(n)
= zk
( n
nk
t
)ψ(nk)
ψ(n)
= αnkzk
( n
nk
t
)
+ βnkzk+1
( n
nk+1
t
)
.
FUNCTIONAL ILL FOR A WIENER PROCESS 25
Hence,
|ξn(t) − z̃nk(t)| ≤
∣∣∣zk(t) − zk
( n
nk
t
)∣∣∣ +
∣∣∣zk+1(t) − zk+1
( n
nk+1
t
)∣∣∣ (17)
Then
sup
n∈[nk,nk+1]
∣∣∣zk(t) − zk
( n
nk
t
)∣∣∣ ≤ sup
s∈[t,ct]
|zk(t) − zk(s)|. (18)
Similarly,
sup
n∈[nk,nk+1]
∣∣∣zk+1(t) − zk+1
( n
nk+1
t
)∣∣∣ ≤ sup
s∈[ t
c ,t]
|zk+1(t) − zk+1(s)|. (19)
For an arbitrary fixed c, we introduce the sets Lδ and Mδ:
Lδ =
{
f(t) ∈ Θ : sup
t≥0
1
1 + t
sup
s∈[t,ct]
|f(s) − f(t)| ≥ δ
}
,
Mδ =
{
f(t) ∈ Θ : sup
t≥0
1
1 + t
sup
s∈[ t
c ,t]
|f(s) − f(t)| ≥ δ
}
From (17)–(19), we get
P
{
sup
t≥0
1
1 + t
sup
n∈[nk,nk+1]
|ξn(t) − z̃nk(t)| ≥ δ
}
≤ P{zk ∈ L δ
2
} + P{zk+1 ∈ M δ
2
}. (20)
The sets Lδ and Mδ are closed in Θ for every c < ∞. We prove this assertion, for
example, only for the set Lδ. Let fn(t) ∈ Lδ and limn→∞ ||fn − f ||Θ = 0. Then the
inequalities
δ ≤ sup
t≥0
1
1 + t
sup
s∈[t,ct]
|fn(s) − fn(t)| ≤ sup
t≥0
1
1 + t
sup
s∈[t,ct]
|fn(s) − f(s)|+
+ sup
t≥0
1
1 + t
sup
s∈[t,ct]
|f(s) − f(t)| + sup
t≥0
1
1 + t
sup
s∈[t,ct]
|fn(t) − fn(t)| ≤
≤ sup
t≥0
1
1 + t
sup
s∈[t,ct]
(1 + s)
|fn(s) − f(s)|
1 + s
+ sup
t≥0
1
1 + t
sup
s∈[t,ct]
|f(s) − f(t)|+
+ ||fn − f ||Θ ≤ c||fn − f ||Θ + sup
t≥0
1
1 + t
sup
s∈[t,ct]
|f(s) − f(t)| + ||fn − f ||Θ
yield
δ ≤ (1 + c)||fn − f ||Θ + sup
t≥0
1
1 + t
sup
s∈[t,ct]
|f(s) − f(t)|.
Passing to the limit in the last inequality as n → ∞, we obtain f ∈ Lδ.
Applying (11), we have
P{zk ∈ L δ
2
} ≤ exp{−ϕ2(nk) inf
f∈Lδ
I(f)}. (21)
Since
sup
s∈[t,ct]
|f(s) − f(t)|2 = sup
s∈[t,ct]
∣∣∣ ∫ s
t
ḟ(u)du
∣∣∣2 ≤ (c − 1)tI(f),
we get
δ2
4
≤ sup
t≥0
1
(1 + t)2
sup
s∈[t,ct]
|f(s) − f(t)|2 ≤ c − 1
4
I(f)
for f ∈ L δ
2
. If we choose c = 1 + 2δ2
R2 , then
inf
f∈Lδ
I(f) ≥ R2
2
. (22)
26 DMITRII S. BUDKOV AND SERGEY YA. MAKHNO
Relations (21) and (22) and the definition of R2 yield
∑
k
P{zk ∈ L δ
2
} ≤
∑
k
exp
{
− ϕ2(nk)
R2
2
}
< ∞. (23)
In an analogous way, we can prove for the same c that∑
k
P{zk+1 ∈ M δ
2
} ≤
∑
k
exp
{
− ϕ2(nk)
R2
2
}
< ∞. (24)
From (20), (23), (24) and the Borel–Cantelli lemma, we get (16). Thus, step 2) is proved.
Step 3). In order to complete the proof of Theorem 1, we need to prove that if R2 ≤ ∞,
then every g ∈ Lr is a limit point of zk for r < R. That is, there exists a sequence nk
such that, with probability 1,
lim
k→∞
ρ(ξnk
, g) = 0. (25)
Applying the Itô formula to the Wiener process w(t) and to the function f(s, x) =
n
n−s(x, a) s ∈ [0, tn
t+n ], n > 0, a is an arbitrary vector from Ed, we have
t + n
n
(
w
( tn
t + n
)
, a
)
=
∫ tn
t+n
0
n
(n − s)2
(w(s), a) ds +
∫ tn
t+n
0
n
n − s
(dw(s), a).
Whence we get∫ tn
t+n
0
n
n − s
dw(s) =
t + n
n
w
( tn
t + n
)
−
∫ tn
t+n
0
n
(n − s)2
w(s) ds =
=
t + n
n
w
( tn
t + n
)
− 1
n
∫ t
0
w
( vn
v + n
)
ds.
(26)
The random process ηn(t) =
∫ tn
t+n
0
n
n−s dw(s) is a process with independent increments,
because it is defined on nonintersecting intervals by independent increments of the Wiener
process. By the property of stochastic integrals, we have
E exp
{
iλ(ηn(t) − ηn(s))
}
= exp
{
− λ2
2
∫ tn
t+n
sn
s+n
n2
(n − u)2
du
}
= exp
{
− λ2
2
(t − s)
}
.
Hence, the process ηn(t) is also a Wiener process for any n. From (26), we obtain
η̃n(t) =
ηn(nt)√
nϕ(n)
= (t + 1)
w
(
n t
t+1
)
√
nϕ(n)
−
∫ t
0
w
(
n s
s+1
)
√
nϕ(n)
ds. (27)
Let a function g ∈ Lr . Define a function f(t) by (9). By virtue of the lemma, f ∈ Kr. It
follows from (8) that the function g(t) admits the representation
g(t) = (t + 1)f
( t
t + 1
)
−
∫ t
0
f
( s
s + 1
)
ds. (28)
From (27) and (28), we get
η̃n(t) − g(t) = (t + 1)
[
w
(
n t
t+1
)
√
nϕ(n)
− f
( t
t + 1
)]
−
∫ t
0
[
w
(
n s
s+1
)
√
nϕ(n)
− f
( s
s + 1
)]
ds.
Whence
|η̃n(t) − g(t)|
1 + t
≤
∣∣∣∣∣
w
(
n t
t+1
)
√
nϕ(n)
− f
( t
t + 1
)∣∣∣∣∣+ t
1 + t
sup
t∈[0,1]
|ξn(t) − f(t)|.
FUNCTIONAL ILL FOR A WIENER PROCESS 27
Thus,
||η̃n − g||Θ ≤ 2||ξn − f ||,
and {
||ξn − f || < δ
}
⊂
{
||η̃n − g||Θ < 2δ
}
. (29)
As has been stated above, the law of the process ηn(t) is the same as that of the process
w(t). Therefore,
P
{
||η̃n − g||Θ < 2δ
}
= P
{
||ξn − g||Θ < 2δ
}
. (30)
Consider the events
Ak = {||ξnk
− f || < δ}, Bk = {||η̃nk
− g||Θ < 2δ}, Ck = {||ξnk
− g||Θ < 2δ}
By the Bulinskii theorem, there exists a sequence nk such that, for the processes ξn(s)
for s ∈ [0, 1] and every δ- neighborhood of the function f :
P{ξnk
∈ (f)δ i.o.} = 1, (31)
where i.o. means ”infinitely often”. Equality (31) yields
P
{ ∞⋂
m=1
∞⋃
l=m
Al
}
= lim
m→∞P
{ ∞⋃
l=m
Al
}
= 1.
By virtue of (29), Ak ⊂ Bk. Hence,
P
{ ∞⋂
m=1
∞⋃
l=m
Bl
}
= lim
m→∞P
{ ∞⋃
l=m
Bl
}
≥ lim
m→∞P
{ ∞⋃
l=m
Al
}
= 1.
For this reason, the events Bk take place infinitely often, and
lim
k→∞
P (Bk) = 1.
¿From this and (30), we get
P
{ ∞⋂
m=1
∞⋃
l=m
Cl
}
= lim
m→∞P
{ ∞⋃
l=m
Cl
}
≥ lim
m→∞P
{
Cm
}
= lim
m→∞P
{
Bm
}
= 1. (32)
It follows from (32) that, with probability 1, the events Ck take place infinitely often.
Equality (25) and Theorem 1 are proved.
3. Small-time functional iterated logarithm law.
We will prove a result similar to Theorem 1 for the process
wn(t) =
w( t
n )√
1
nϕ(n)
, t ∈ [0,∞), n ≥ 3. (33)
To this end, we make use a method from [3]. On the space Θ, we define a time inversion
transformation T by the formula
(Tθ)(t) =
⎧⎨
⎩
0, t = 0,
t θ(
1
t
), t > 0.
In [3], it is noted that the transformation T is bijective on Θ. Furthermore, T is an
isometry from Θ onto Θ and from H1 onto H1:
||θ||Θ = ||Tθ||Θ, ||θ||H1 = ||Tθ||H1 . (34)
Theorem 2. For ϕ ∈ Φ, the set of limit points of sequence (33) is LR with probability
1, where R is defined in (5).
28 DMITRII S. BUDKOV AND SERGEY YA. MAKHNO
Proof. We denote
w̃(t) = (Tw)(t) =
⎧⎨
⎩
0, t = 0,
t w(
1
t
), t > 0.
It is known that w̃(t) is also a Wiener process. For t > 0,
w̃(nt)√
nϕ(n)
= t
w(1
t
1
n )√
1
nϕ(n)
. (35)
From (35), we get
sup
t≥0
1
1 + t
∣∣∣∣∣ w̃(nt)√
nϕ(n)
− (Tf)(t)
∣∣∣∣∣= sup
t≥0
1
1 + t
∣∣∣∣∣wn(t) − f(t)
∣∣∣∣∣. (36)
Suppose that there exists a subsequence nk such that
lim
nk→∞ ||wnk
− f ||Θ = 0. (37)
From (36), we have
lim
nk→∞ sup
t≥0
1
1 + t
∣∣∣∣∣ w̃(nkt)√
nϕ(nk)
− (Tf)(t)
∣∣∣∣∣= 0. (38)
According to Theorem 1, we conclude that (Tf)(t) ∈ LR. Then relations (34) imply that
f(t) ∈ LR. Thus, each limit point of the sequence wn(t) is an element of LR. Conversely,
let a function f(t) ∈ LR. Then, by (34), the function (Tf)(t) ∈ LR and, by virtue of
Theorem 1, there exists a subsequence nk such that
lim
nk→∞ sup
t≥0
1
1 + t
∣∣∣∣∣ w̃(nkt)√
nϕ(nk)
− (Tf)(t)
∣∣∣∣∣= 0.
This relation and (36) yield (37). Theorem 2 is proved.
Bibliography
1. A.V. Bulinskii, Large deviations of diffusion processes with discontinuous drift and their occu-
pation times, Theory Appl. Probab 19 (1980), 509-524.
2. J.D. Deuschel, D.W. Stroock, Large Deviations, Academic Press, Boston, 1989.
3. N. Gantert, An inversion of Strassen’s law of the iterated logarithm for small time, Annals of
Probability. 21(2) (1993), 1045-1049.
E-mail : budkov@iamm.ac.donetsk.ua
E-mail : makhno@iamm.ac.donetsk.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4503 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T13:22:58Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Budkov, D.S. Makhno, S.Ya. 2009-11-19T13:57:42Z 2009-11-19T13:57:42Z 2007 Functional iterated logarithm law for a Wiener process / D.S. Budkov, S.Ya. Makhno // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 3. — С. 22–28. — Бібліогр.:3 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4503 519.21 The functional iterated logarithm law for a Wiener process in the Bulinskii form for great and small times is proved. en Інститут математики НАН України Functional iterated logarithm law for a Wiener process Article published earlier |
| spellingShingle | Functional iterated logarithm law for a Wiener process Budkov, D.S. Makhno, S.Ya. |
| title | Functional iterated logarithm law for a Wiener process |
| title_full | Functional iterated logarithm law for a Wiener process |
| title_fullStr | Functional iterated logarithm law for a Wiener process |
| title_full_unstemmed | Functional iterated logarithm law for a Wiener process |
| title_short | Functional iterated logarithm law for a Wiener process |
| title_sort | functional iterated logarithm law for a wiener process |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4503 |
| work_keys_str_mv | AT budkovds functionaliteratedlogarithmlawforawienerprocess AT makhnosya functionaliteratedlogarithmlawforawienerprocess |