Statistics & Probability Letters 53 (2001) 21 – 26
The strong approximation for the Kesten–Spitzer
random walk Li-Xin Zhang
Department of Mathematics, Zhejiang University, Xixi Campus, Zhejiang, Hangzhou 310028, China
Received January 2000; received in revised form September 2000
Abstract
In this note we study the strong approximation for a one-dimensional simple random walk in a general i.i.d. scenery
when the scenery has only 0nite lower moments. Namely, an approximation is obtained when the scenery has only 0nite
c 2001 Elsevier Science B.V. All rights reserved
(2 + )th moments. MSC: 60F15; 60J55; 60J15; 60J65
Keywords: Local time; Random walk in random scenery; Brownian motion in Brownian scenery; Strong approximation
Let = {x }x∈Z (sometimes also written {(x)}x∈Z ) denote a sequence of independent and identically
distributed real-valued random variables (i.i.d. r.v.’s) such that
E(0 ) = 0
and
E(02 ) = 1:
(1)
Any realization of the sequence {x }x∈Z is called a “scenery”. Let S = {Sk }k∈N0 be a simple symmetric
random walk on Z starting at S0 = 0, independent of . The process K = {K(n)}n∈N0 , de0ned by
K(n) =
n
(Sk );
n ∈ N0
(2)
k=o
is usually referred to as the Kesten–Spitzer random walk in random scenery (cf. Kesten and Spitzer, 1979,
R@ev@esz, 1990).
There is a continuous analogue for K introduced and analyzed by Kesten and Spitzer (1979). To describe
this, let B = {B(t); t ¿ 0} and W = {W (x); x ∈ R} be independent real-valued standard Brownian motions
with B(0) = W (0) = 0. Let {L(t; x); t ¿ 0; x ∈ R} denote the jointly continuous version of the local time
Research supported by National Natural Science Foundation of China (No. 10071072).
E-mail address: lxzhang@mail.hz.zj.cn (L.-X. Zhang).
c 2001 Elsevier Science B.V. All rights reserved
0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 0 ) 0 0 2 4 3 - 1
22
L.-X. Zhang / Statistics & Probability Letters 53 (2001) 21 – 26
process of B. Now, de0ne the process G, which is called Brownian motion in Brownian scenery, by
G(t) = L(t; x) dW (x); t ¿ 0:
(3)
R
It is proved by Kesten and Spitzer (1979) that
in law
{n−3=4 K([nt]); 0 6 t 6 1} → {G(t); 0 6 t 6 1};
(4)
in law
where “ → ” stands for weak convergence in D[0; 1]. Khoshnevisan and Lewis (1998) studied a strong vision
of (4) in the special case where the random scenery is Gaussian. More precisely, they proved that, if 0 is
a Gaussian N(0; 1) variables, then (on a suitably enlarged probability space) one can construct a pair of K
and G such that, for any ¿ 0,
a:s:
max |K(m) − G(m)| = o(n1=2+ );
06m6n
n → ∞:
(5)
Cs@aki et al. (1999) extended (5) to a more general random scenery. They proved the following theorem.
Theorem A. Let {x }x∈Z be a random scenery satisfying (1) and E|0 |p ¡ ∞ for all p ¿ 0. Then on a
suitably enlarged probability space there exists a coupling of K and G; such that; for any ¿ 0;
a:s:
max |K(m) − G(m)| =o(n5=8+ );
06m6n
n → ∞:
(6)
An application of Theorem A yields the law of iterated logarithm:
lim sup
n→∞
K(n)
25=4
a:s:
=
3=4
(n log log n)
3
(7)
if {x }x∈Z satis0es (1) and E|0 |p ¡ ∞; for all p ¿ 0.
The aim of this paper is to show (6) and (7) under the conditions of 0nite lower absolute moments. Our
result reads as follows:
Theorem 1. Let {x }x∈Z be a random scenery satisfying (1) and E|0 |p ¡ ∞; where 2 6 p 6 4. Then on
a suitably enlarged probability space there exists a coupling of K and G; such that; for any ¿ 0,
a:s:
max |K(m) − G(m)| =o(n1=2+1=(2p)+ );
06m6n
n → ∞:
(8)
An application of Theorem 1 yields that (7) holds if E|0 |2+ ¡ ∞ for some ¿ 0.
Denote the number of walker’s visits to x until time n by
(n; x) =
n
I {Sk = x};
n ∈ N 0 ; x ∈ Z;
(9)
k=0
which is often referred to as the local time of the random walk S. Then (2) can be rewritten as
x (n; x); n ∈ N0 :
K(n) =
x∈Z
We shall prove Theorem 1 in two steps.
(10)
L.-X. Zhang / Statistics & Probability Letters 53 (2001) 21 – 26
23
Step 1. Let = {x }x∈Z satisfy (1). Then there is a coupling of , S and B such that is independent of
(S; B) and for any ¿ 0,
a:s:
x ((n; x) − L(n; x)) =o(n1=2+ ); n → ∞:
(11)
x∈Z
Proof. According to a theorem by R@ev@esz (1981) and a result by Khoshnevisan and Lewis (1998), one can
construct a random walk S from the Brownian motion B such that, for every ¿ 0,
a:s:
sup|(n; x) − L(n; x)| =O(n1=4+ );
n → ∞;
x∈Z
(12)
and for any q ¿ 1,
supE(|(n; x) − L(n; x)|q ) 6 cq nq=4 ;
x∈Z
n ∈ N;
(13)
where cq is a constant depending only on q.
De0ne
I (N; n) =
N
x ((n; x) − L(n; x));
N; n ∈ N:
(14)
x=−N
Following the argument of Cs@aki et al. (1999), in order to prove (11) we only need to show that, for every
¿ 0,
a:s:
I ([n1=2+ ]; n) =O(n1=2+ ):
(15)
For each x ∈ Z, let Xx = x I {|x | ¿ |x|1=2 }; Yx = x I {|x | 6 |x|1=2 } − Ex I {|x | 6 |x|1=2 } and YP x =
x I {|x | ¿ |x|1=2 } − Ex I {|x | ¿ |x|1=2 }. De0ne
I1 (N; n) =
N
Xx ((n; x) − L(n; x));
x=−N
I2 (N; n) =
N
Yx ((n; x) − L(n; x));
x=−N
I3 (N; n) =
N
((n; x) − L(n; x))Ex I {|x | 6 |x|1=2 };
x=−N
for all N; n ∈ N. Note that
∞
P(|x | ¿ |x|1=2 ) 6 cE02 ¡ ∞:
x=−∞
It follows that
P(Xx = 0; i:o:) = 0:
On the other hand, by (12), for each 0xed x ∈ Z,
a:s:
Xx ((n; x) − L(n; x)) =o(n1=4+ );
n → ∞;
(16)
24
L.-X. Zhang / Statistics & Probability Letters 53 (2001) 21 – 26
and therefore
a:s:
I1 ([n1=2+ ]; n) = o(n1=4+ );
n → ∞:
(17)
For I3 , by (12) we have
N
1=2 |I3 (N; n)| = ((n; x) − L(n; x))Ex I {|x | ¿ |x| }
x=−N
6
N
|(n; x) − L(n; x)|E|0 |I {|0 | ¿ |x|1=2 }
x=−N

N
6 sup|(n; x) − L(n; x)| E|0 | +
x

|x|−1=2 E02 
x=−N; x=0
a:s:
=O(N 1=2 )o(n1=4+=2 ) as n → ∞; N → ∞:
Hence
a:s:
I3 ([n1=2+ ]; n) =O(n1=4+=2 )o(n1=4+=2 ) = o(n1=2+ );
n → ∞:
(18)
At last, by (16) – (18), it suRces to show the summability of P(I2 ([n1=2+ ]; n) ¿ n1=2+ ). For 0xed (S; B),
by the Rosenthal inequality (cf., Petrov, 1995) we have for any q ¿ 2,


q=2
N
N
 
E[|I2 (N; n)|q |(S; B)] 6 cq
((n; x) − L(n; x))2 EYx2
+
|(n; x) − L(n; x)|q E|Yx |q


x=−N
6 cq

N
 
x=−N
q=2
((n; x) − L(n; x))2 Ex2
+
x=−N
6 cq (2N + 1)
N
x=−N
q=2−1
N
|(n; x) − L(n; x)|q N (q−2)=2 Ex2



|(n; x) − L(n; x)|q
x=−N
6 cq N q=2−1
N
|(n; x) − L(n; x)|q :
(19)
x=−N
By (13) and (19), we have
E|I2 (N; n)|q 6 cq N q=2−1
N
E|(n; x) − L(n; x)|q 6 cq N q=2 nq=4 :
x=−N
Hence
P(|I2 ([n1=2+ ]; n) ¿ n1=2+ ) 6 cq n−(1=2+)q n(1=2+)q=2 nq=4 = cq n−q=2 ;
which is summable for any q ¿ 2=. This concludes the proof of Step 1.
Step 2. Let {x }x∈Z satisfy (1) and E|0 |p ¡ ∞, where 2 6 p 6 4. Then there is a coupling of ; S
and W such that (; W ) is independent of (S; B) and for any ¿ 0,
a:s:
x L(n; x) − L(n; x) dW (x) = o(n1=2+1=(2p)+ ); n → ∞:
(20)
x∈Z
R
L.-X. Zhang / Statistics & Probability Letters 53 (2001) 21 – 26
25
Proof. Following the line of the proof of Proposition 2:2 of Cs@aki et al. (1999), it suRces to show that, for
any 0xed b ¿ 12 and any ¿ 0,
a:s:
J ([nb ]; n) = o(n1=4+b(1=2+1=p)+ );
where
J (N; n) = Wn
!(N )
0
{Wn (t); t ¿ 1}
n → ∞;
(21)
A2n (x) d x
;
N; n ∈ N;
(22)
is a Brownian motion for each n ∈ N;
An (x) = L(n; x) − L(n; j) if x ∈ (!j−1 ; !j ];
!(n) = !n =
n
Ti ;
n∈N
i=1
p=2
and {Ti }∞
6
i=1 is a sequence of i.i.d. non-negative random variables with ET1 = Var(0 ) = 1 and ETi
p
cp E|0 | ¡ ∞.
By (2:21) and (2:22) of Cs@aki et al. (1999) we have for any a ¿ 0 that
a:s:
max max |Wi (s)| =O(na=2 (log n)1=2 );
16i6n 06s6na
a:s:
sup |L(t; x) − L(t; y)| =o(t 1=4+a=2+ );
|x−y|6t a
n → ∞;
(23)
t → ∞; ∀ ¿ 0:
(24)
On the other hand, since the sequence {Ti }i∈N is i.i.d. with ETi = 1; E|Ti |p=2 ¡ ∞ and 1 6 p=2 6 2, by the
classical LLN or LIL, we obtain
if 2 6 p ¡ 4;
n + o(n2=p )
a:s:
!(n) =
n → ∞:
(25)
n + O((n log log n)1=2 ) if p = 4;
Now 0x b ¿ 12 and ¿ 0. For any 0 ¡ x 6 !([nb ]), there exists 1 6 j 6 [nb ] such that x ∈ (!j−1 ; !j ].
Since by (25), |x − j| 6 n2b=p+ and x 6 2nb , we can apply (24) to conclude that
sup
a:s:
|An (x)| = o(n1=4+b=p+ );
06x6!([nb ])
Therefore, by (25),
!([nb ])
A2n (x) d x 6 !([nb ])
0
sup
n → ∞:
a:s:
A2n (x) = o(nb+1=2+2b=p+2 ):
06x6!([nb ])
(26)
By (23) and (26), we have
a:s:
J ([nb ]; n) = O(n1=4+b(1=2+1=p)+ (log n)1=2 ) = o(n1=4+b(1=2+1=p)+2 ):
The proof of Step 2 is now completed.
Putting together (10), (11) and (20), we get (8).
References
Cs@aki, E., KSonig, W., Shi, Z., 1999. An embedding of the Kesten–Spitzer random walk in random scenery. Stochastic Process. Appl. 82,
283–292.
Kesten, H., Spitzer, F., 1979. A limit theorem related to a new class of self similar processes. Z. Wahrsch. Verw. Gebiete 50, 5–25.
26
L.-X. Zhang / Statistics & Probability Letters 53 (2001) 21 – 26
Khoshnevisan, D., Lewis, T.M., 1998. A law of the iterated logarithm for stable processes in random scenery. Stoch. Process. Appl. 74,
89–121.
Petrov, V.V., 1995. Limit Theorems of Probability Theory. Oxford Science Publications, Oxford.
R@ev@esz, P., 1981. Local time and invariance. In: Dugu@e et al. (Eds.), Analytical Methods in Probability Theory, Lecture Notes in
Mathematics, Vol. 861, Springer, Berlin, pp. 128–145.
R@ev@esz, P., 1990. Random Walk in Random and Non-Random Environments. World Scienti0c, Singapore.