Article11 Sghir Aissa .pdf



Nom original: Article11_Sghir_Aissa.pdf
Titre: LSTM_A_1015571_O

Ce document au format PDF 1.4 a été généré par dvips(k) 5.95a Copyright 2005 Radical Eye Software / iText 4.2.0 by 1T3XT, et a été envoyé sur fichier-pdf.fr le 17/02/2017 à 20:32, depuis l'adresse IP 105.145.x.x. La présente page de téléchargement du fichier a été vue 194 fois.
Taille du document: 97 Ko (12 pages).
Confidentialité: fichier public




Télécharger le fichier (PDF)










Aperçu du document


This article was downloaded by: [New York University]
On: 02 July 2015, At: 05:03
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
office: 5 Howick Place, London, SW1P 1WG

Stochastic Models
Publication details, including instructions for authors and
subscription information:
http://www.tandfonline.com/loi/lstm20

A Law of the Iterated Logarithm
for Some Additive Functionals of
Symmetric Stable Process Via the Strong
Approximation
a

M. Ait Ouahra & A. Sghir

a

a

Faculté des Sciences Oujda, Laboratoire de Modélisation
Stochastique et Déterministe et URAC (04), Oujda, Morocco
Published online: 22 Jun 2015.

Click for updates
To cite this article: M. Ait Ouahra & A. Sghir (2015): A Law of the Iterated Logarithm for Some
Additive Functionals of Symmetric Stable Process Via the Strong Approximation, Stochastic Models,
DOI: 10.1080/15326349.2015.1015571
To link to this article: http://dx.doi.org/10.1080/15326349.2015.1015571

PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the
“Content”) contained in the publications on our platform. However, Taylor & Francis,
our agents, and our licensors make no representations or warranties whatsoever as to
the accuracy, completeness, or suitability for any purpose of the Content. Any opinions
and views expressed in this publication are the opinions and views of the authors,
and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content
should not be relied upon and should be independently verified with primary sources
of information. Taylor and Francis shall not be liable for any losses, actions, claims,
proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or
howsoever caused arising directly or indirectly in connection with, in relation to or arising
out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any
substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,
systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

Downloaded by [New York University] at 05:03 02 July 2015

Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions

Stochastic Models, 00:1–10, 2015
C Taylor & Francis Group, LLC
Copyright
ISSN: 1532-6349 print / 1532-4214 online
DOI: 10.1080/15326349.2015.1015571

Downloaded by [New York University] at 05:03 02 July 2015

A LAW OF THE ITERATED LOGARITHM FOR SOME ADDITIVE
FUNCTIONALS OF SYMMETRIC STABLE PROCESS VIA THE STRONG
APPROXIMATION

M. Ait Ouahra and A. Sghir
Facult´e des Sciences Oujda, Laboratoire de Mod´elisation Stochastique et D´eterministe et
URAC (04), Oujda, Morocco
2

In this note we give a strong approximation version for the first-order limit theorem of symmetric
stable process of index 1 < α ≤ 2. This generalizes a result of Csaki et al.[7] for Brownian motion.
As an application, we give the law of iterated logarithm for some additive functionals of this process.
Keywords Additive functional; Barlow-Yor inequality; Law of iterated logarithm; Local
time; Stable process; Strong approximation.
Mathematics Subject Classification

60F15; 60J55; 60G18.

1. INTRODUCTION
In this article, we are interested in the strong approximation and law of
iterated logarithm of a class of continuous additive functionals of symmetric
stable process of index 1 < α ≤ 2.
Throughout this note X := ( , F, Ft , θt , Xt , P x ) will denote the canonical realization of a real valued symmetric stable process of index 1 < α ≤ 2,
with X0 = 0. The sample paths of Xt are right-continuous with left limits almost surely (c`adl`ag for brevity), and has stationary independent increments
with characteristic function:
E0 exp(iλXt ) = exp(−t|λ|α ), t ≥ 0, λ ∈ R.
E0 denotes the expectation with respect to the distribution P 0 of the process
starting from 0, and (θt ): → are the translation operators defined by
(θt (ω))(s ) = ω(t + s ).
Notice that for α = 2, X is a Brownian motion.
Received June 2014; Accepted February 2015
Address correspondence to M. Ait Ouahra, Facult´e des Sciences Oujda, Laboratoire de Mod´elisation
Stochastique et D´eterministe et URAC Ø4 B.P. 717, Oujda 6000, Morocco; E-mail: m.aitouahra@ump.ma

2

Ait Ouahra and Sghir

For all 0 ≤ t ≤ T , we define the random measure μt (.) by


t

μt (A) =

1A (Xs )ds,

Downloaded by [New York University] at 05:03 02 July 2015

0

where A ⊂ R is a Borel set of R and 1A (.) is its characteristic function. The
measure μt (A) is called the occupation measure of X on the Borel set A. It
is well known by Barlow[2], Blumenthal and Getoor[4] and Boylan[5] that the
measure μt (A) has a density, denoted by Ltx , with respect to the Lebesgue
measure, and {Ltx , t ≥ 0, x ∈ R} is called the family of local times associated
with X . Moreover, Ltx has a version that is almost surely continuous as well as
uniformly continuous on any compact set, and satisfies the so-called scaling
property:
1
α−1 xλ α1
x
, t ≥ 0, x ∈ R, P y λ α ∀λ > 0,
Lt , t ≥ 0, x ∈ R, P y d λ− α Lλt

(1)

where d means the equality in the sense of the finite-dimensional distributions.
In addition, Ltx is an additive functional,
x
(θs (ω)),
Ltx (ω) = Lsx (ω) + Lt−s

(2)

and satisfies the occupation density formula:


t


f (Xs (ω))ds =

0

R

f (x)Ltx (ω)d x,

(3)

for any bounded or nonnegative Borel function f .
Most of the estimates in this article contain unspecified positive constants. We use the same symbol C for these constants, even when they vary
from one line to the next.
By Lemma 3.3 in Marcus and Rosen[13] and Theorem 1 in Ait Ouahra
and Eddahbi[1] and the Kolmogorov criterion, for any fixed T > 0, we have
almost surely:
, ∃ 0 < C < ∞ such that ∀ 0 ≤ t, s ≤ T , |x| ≤ M
∀ 0 < β < α−1
α
|Ltx − Lsx | ≤ C|t − s |β .
∀ 0 < β1 <
|x|, |y | ≤ M

α−1
,


∀ 0 < β2 <

α−1
,
2

∃ 0 < C < ∞ such that ∀ 0 ≤ t, s ≤ T ,

|Ltx − Lt − Lsx + Lsy | ≤ C|t − s |β1 |x − y |β2 ,
y

(4)

(5)

Law of the Iterated Logarithm

3

where M is a finite positive constant.
The following lemma is an easy consequence of the scaling property (1).
It will be useful in the proof of our main result.
Lemma 1.1.



x
0
− α−1
x
0
sup Lt , t ≥ 0, P d λ α sup Lλt , t ≥ 0, P
x∈R

For any 0 < ν <

α−1
,
2

Downloaded by [New York University] at 05:03 02 July 2015



x∈R

(6)

we have



y
|Lsx − Ls |
0
sup sup
, t ≥ 0, P
ν
0≤s ≤t x = y |x − y |



y
x
|Lλs
− Lλs |
ν
− α−1
+
0
, t ≥ 0, P
d λ α α sup sup
ν
0≤s ≤t x = y |x − y |

(7)

Remark 1.1. For more applications of the scaling property (1), we refer to
Chen et al.[6] and Fitzsimmons and Getoor[10].

Let f be a Borel function on R such that ¯f := R f (x)d x = 0. The first
order limit theorem of Darling and Kac[8] states that the family of processes,


1



λt

1

λ2


f (Bs )ds, t ≥ 0

0

converges in law, in the space of continuous functions, as λ goes to infinity,


to the process { f l t0 , t > 0}, where B is a Brownian motion and l tx its local
time. A strong approximation version of this result is given in Csaki et al.[7]:
For all sufficiently small ε > 0, when t goes to infinity, we have


t
0

1
f (Bs )ds = ¯f l t0 + o t 2 −ε , a.s.



with R |x|k f (x)d x < ∞ for some k > 0.
By an extension (see Kasahara[11]), of a famous theorem of Darling


and Kac[8], if f ∈ L 1 (R) such that f = R f (x)d x = 0, then the family of
processes,


1
λ

α−1
α



λt
0


f (Xs )ds, t ≥ 0

4

Ait Ouahra and Sghir

converges in law, in the space of continuous functions, as λ goes to infinity,


Downloaded by [New York University] at 05:03 02 July 2015

to the process { f Lt0 , t ≥ 0}.
Our aim in this article is to give a strong approximation analogue of this
limit theorem. As
tan application, we deduce the law of the iterated logarithm
of the process { 0 f (Xs )ds, t ≥ 0}.
We briefly explain the structure of the rest of the article. In the next
section we state and prove the main result. In the last section we deduce
the law of the iterated logarithm of some additive functionals of symmetric
stable process of index 1 < α ≤ 2.

2. STRONG APPROXIMATION
The main result of this article is the following.


Theorem 2.1. Let f be a Borel function on R such that f =


R

f (x)d x = 0 and


R

|x|k | f (x)|d x < ∞,

(8)

for some k > 0. Then for all sufficiently small ε > 0, when t goes to infinity, we have


t
0



f (Xs )ds = f Lt0 + o(t

α−1
α −ε

),

a.s.

(9)

To prove this theorem, we need the following lemma (see Fitzsimmons
and Getoor[10], page 326).
Lemma 2.1. Let (At )t≥0 be a continuous increasing (Ft )-adapted real valued process with A0 = 0. Assume that
(i) At ≤ As + K.At−s ◦ θs , ∀ t, s ≥ 0, for some constant K > 0.
(ii) There exists a constant q > 0, such that


1
lim sup P y Aλ > λ q z = 0.
z→∞ λ>0,y ∈R

Then for each p > 0, there exists a constant 0 < C < ∞, such that
1

At p ≤ Ct q ,
1

where . p = (E0 |.|p ) p .

∀t ≥ 0,

5

Law of the Iterated Logarithm

As an application of Lemma 2.1, we obtain the following inequality, which
generalizes the well-known Barlow-Yor inequality for Brownian motion to
symmetric stable process of index 1 < α ≤ 2 (see Barlow and Yor[3] for the
Brownian motion case).
Lemma 2.2. For all t > 0, 0 < ν <
C < ∞, such that

α−1
2

and p ≥ 1, there exists a constant 0 <


x


L − Lsy
α−1
ν


s
≤ Ct α − α .
sup sup
ν
0≤s ≤t x = y |x − y |
Downloaded by [New York University] at 05:03 02 July 2015

p

Proof. We apply Lemma 2.1 with
y

|L x − Ls |
At = sup sup s
,q =
ν
0≤s ≤t x = y |x − y |



α−1 ν

α
α

−1

.

Clearly, (At ) is increasing and satisfies (i) of Lemma 2.1.
In fact, for all r ≤ s ≤ t, we have
y

y

y
|Lrx + Lsx−r ◦ θr − Lr − Ls −r ◦ θr |
|L x − Ls |
=
sup
sup s
ν
|x − y |ν
x = y |x − y |
x = y
y

≤ sup
x = y

y
|Lsx−r ◦ θr − Ls −r ◦ θr |
|Lrx − Lr |
+
sup
|x − y |ν
|x − y |ν
x = y
y

≤ sup sup
0≤v≤r x = y

y
|Lsx−r ◦ θr − Ls −r ◦ θr |
|Lvx − Lv |
+
sup
|x − y |ν
|x − y |ν
x = y
y

|L x − Ls −r |
= Ar + sup s −r
◦ θr .
|x − y |ν
x = y
Therefore,
y

sup sup

0≤s ≤t x = y

y
|Lsx−r − Ls −r |
|Lsx − Ls |

A
+
sup
sup
◦ θr
r
|x − y |ν
|x − y |ν
0≤s ≤t x = y
y

|L x − Ls −r |
≤ Ar + sup sup s −r
◦ θr
|x − y |ν
0≤s −r ≤t−r x = y
y

|L x − Lv |
= Ar + sup sup v
◦ θr .
ν
0≤v≤t−r x = y |x − y |

6

Ait Ouahra and Sghir

Finally,
At ≤ Ar + At−r ◦ θr .
It follows from (5) that
0 ≤ At − As ≤ At−s ◦ θs ≤ C sup |r − s |β1 ≤ C|t − s |β1 ,

Downloaded by [New York University] at 05:03 02 July 2015

s ≤r ≤t

for any 0 < β1 < α−1
, which implies that At is continuous.

x−y
x
Since Lt (τ y (ω)) = Lt (ω), where τ y : → is the translation ω →
ω(.) + y , and X is spatially homogenous, i.e., (τ y (P 0 ) = P y ), it follows that
the P y distribution of At does not depend on y. Thus, by (7) we have
lim

1

1

sup P y (Aλ > λ q z) = lim sup P 0 (Aλ > λ q z)

z→∞ λ>0,y ∈R

z→∞ λ>0

= lim P 0 (A1 > z) = 0,
z→∞

where we have used in the last estimation the fact that (5) and L0x = 0, implies
y

A1 = sup sup
0≤s ≤1 x = y

|Lsx − Ls |
< ∞,
|x − y |ν

a.s.


This completes the proof of Lemma 2.2.
α−1
2

Corollary 2.1. For any 0 < ν <

and ε > 0, when t goes to infinity, we have

α−1 ν
|L x − Lt |
=
o
t α − α +ε ,
sup t
ν
|x

y
|
x = y
y

a.s.

(10)

Proof. Using Tchebychev’s inequality and Lemma 2.2 with p = 2ε , for any
n ≥ 1, we have

P0

y

|L x − Ls |
α−1
ν
> n α − α +ε
sup sup s
ν
0≤s ≤n x = y |x − y |



≤ C(ν, ε)n−2 .

Then, by the Borel-Cantelli lemma, we get as n → +∞, almost surely,
α−1 ν
|L x − Ls |
=
O
n α − α +ε .
sup sup s
ν
0≤s ≤n x = y |x − y |
y

Law of the Iterated Logarithm

7

Since At is increasing and ε can be arbitrarily small, we have proved the

corollary.
We need also the following lemma.
Lemma 2.3. For any ε > 0, when t goes to infinity, we have
α−1
sup Ltx = o t α +ε ,

Downloaded by [New York University] at 05:03 02 July 2015

x∈R

a.s.

(11)

Proof. This lemma follows immediately by the same arguments used in the
proof of Corollary 2.1. More precisely, we apply Lemma 2.1 with
At = sup Ltx = sup sup Lsx , q =
0≤s ≤t x∈R

x∈R

α−1
,
α


jointly with (2), (4), and (6).

We are now able to prove Theorem 2.1. The idea of the proof is inspired
from that used in Csaki et al.[7] for the Brownian motion case.
Proof of Theorem 2.1. Our aim is to estimate


t

I (t) =
0



f (Xs )ds − f Lt0 .

Without loss of generality, we suppose that f is a real positive function (in
the general case we use the following decomposition f = f + − f − ).
By the occupation density formula (3), we have

I (t) =

R



f (x) Ltx − Lt0 d x := I1 (t) + I2 (t),

where

I1 (t) =

|x|>t a

f (x)(Ltx − Lt0 )d x,


I2 (t) =
for some 0 < a < 1.

|x|≤t a

f (x)(Ltx − Lt0 )d x,

8

Ait Ouahra and Sghir

Let us deal with the first term I1 (t). By (8) and (11), we have

|I1 (t)| ≤ sup

|x|>t a

|Ltx



Lt0 |

|x|>t a

α−1

|x|−k |x|k | f (x)|d x = o t α −ak+ε . a.s.

Now, we deal with I2 (t). Using (10) and the fact that f is integrable, we
obtain

Downloaded by [New York University] at 05:03 02 July 2015

|I2 (t)| ≤ t



|L x − L 0 |
sup t ν t
|x|
|x|≤t a


|x|≤t a

α−1 ν

| f (x)|d x = o t α − α +aν+ε . a.s.

Therefore
|I (t)| = o(t

α−1
ν
α − α +aν+ε

) + o(t

α−1
α −ak+ε

). a.s.

Choosing
a=

ν
,
α(ν + k)

it is clear that 0 < a < 1. It follows that
|I (t)| = o(t b+ε ),

a.s.

with
b=

ν
α−1
α−1
−k
<
.
α
α(ν + k)
α

Then for all sufficiently small ε > 0, when t goes to infinity,
I (t) = o(t

α−1
α −ε

),

The theorem is proved.

a.s.


3. LAW OF ITERATED LOGARITHM
An immediate application of our strong approximation result is to
obain
the law of iterated logarithm, (LIL for brevity), for the process
t
{ 0 f (Xs )ds, t ≥ 0}.

9

Law of the Iterated Logarithm

It is well known by Kesten[12] that the local time of the Brownian motion
satisfies the following LIL,
supl tx

l t0

x∈R

lim sup
= lim sup
= 1.
2t log log t
2t log log t
t→∞
t→∞

a.s.

A similar result for local time of symmetric stable process is given by Donsker
and Varadhan[9] (Example 5, page 752), more precisely,

Downloaded by [New York University] at 05:03 02 July 2015

lim sup
t→∞

supLtx

Lt0
t

α−1
α

(log log t)

= lim sup

1
α

t→∞

x∈R

t

α−1
α

1

(log log t) α

= d(α),

a.s.

where
d(α) =

( α1 ) (1 − α1 )
1

π (α − 1)1− α

.

By using (9), we obtain immediately the following result.


Theorem
3.1. Let f be a Borel function on R such that f =

k
|x|
|
f
(x)|d
x < ∞ for some k > 0. Then,
R
t
lim sup
t→∞

0

t

α−1
α



f (Xs )ds

(log log t)

1
α

= f d(α).


R

f (x)d x = 0 and

a.s.

Remark 3.1. If we can prove the similar result condition (ii) in Lemma 2.1
for fractional Brownian motion, our results remain valid for this case.
ACKNOWLEDGMENTS
The authors would like to thank the anonymous referee for her/his
careful reading of the manuscript and useful comments.
REFERENCES
1. M. Ait Ouahra; M. Eddahbi. Th´eor`emes limites pour certaines fonctionnelles associ´ees aux processus
stable sur l’espace de H¨older. Publ. Math. 2001, 45, 371–386.
2. M. T. Barlow. Necessary and sufficient conditions for the continuity of local times of L´evy processes.
Ann. Prob. 1988, 16, 1389–1427.
3. M. T. Barlow; M. Yor. Semi-martingale inequalities via the Garcia-Rodemich-Rumsey lemma, and
applications to local times. J. Funct. Anal. 1982, 49, 198–229.

Downloaded by [New York University] at 05:03 02 July 2015

10

Ait Ouahra and Sghir

4. R. M. Blumenthal; R. K. Getoor. Markov processes and potential theory. Academic. Press: New York, 1968.
5. E. S. Boylan. Local times for a class of Markov processes. Illinois J. Math. 1964, 8, 19–39.
6. X. Chen; W. V. Li; J. Rosen. Large deviations for local times of stable processes and stable random
walks in 1 dimension. Electron. J. Probab. 2005, 10, 577–608.
} A. F¨oldes; P. R´ev´esz. Strong approximation of additive functionals. J. Theoretical.
7. E. Csaki; M. Cs¨orgo;
Probab. 1992, 5, 679–706.
8. D. A. Darling; R. K. Kac. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 1957,
84, 444–458.
9. M. D. Donsker; S. R. S. Varadhan. On laws of the iterated logarithm for local times. Comm. Pur. Appl.
Math. 1977, 30, 707–753.
10. P. J. Fitzsimmons; R. K. Getoor. Limit theorems and variation properties for fractional derivatives of
the local time of a stable process. Ann. Inst. H. Poincar´e. 1992, 28, 311–333.
11. Y. Kasahara. Limit theorems of occupation times for Markov process. Publ. RIMS Kyoto Univ. 1977,
12, 801–818.
12. H. Kesten. An iterated logarithm law for local time. Duke. Math. J. 1965, 32, 447–456.
13. M. B. Marcus; J. Rosen. p-variation of the local times of symmetric stable processes and of Gaussian
processes with stationary increments. Ann. Probab. 1992, 20, 1685–1713.



Documents similaires


article11 sghir aissa
article3 sghir aissa
article4 sghir aissa
article14 sghir aissa
article8 sghir aissa
article15 sghir aissa


Sur le même sujet..