Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Copyright © 2018 Wensheng Wang and Anwei Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let X = {Xi, i ≥ 1} be a sequence of real valued random variables, S0 = 0 and Sk = ∑ki=1 Xi (k ≥ 1). Let σ = {σ(x), x ∈ Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn = ∑nk=0 σ(⌊Sk⌋) (n ≥ 0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋ ≤ a < ⌊a⌋ + 1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0 < β < 2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random sceneryKn. The obtained results supplement to some corresponding results in the literature.


Introduction
Let  = {  ,  ≥ 1} be a sequence of real valued random variables,  0 = 0 and   = ∑  =1   ( ≥ 1).Let  = {(),  ∈ Z} be a sequence of R-valued random variables which are independent of 's.We refer to  = {  , ≥ 0} as the random walk and  as the random scenery.Then the process  = {  ,  ∈ N} is defined by where N = {0, 1, 2, . ..} and ⌊⌋ means the unique integer satisfying ⌊⌋ ≤  < ⌊⌋ + 1, called a random walk in random scenery (RWRS, in short), sometimes also referred to as the Kesten-Spitzer random walk in random scenery; see Kesten and Spitzer [1].An interpretation is as follows.If a random walker has to pay () units at any time he/she visits the site , then   is the total amount he/she pays by time .RWRS was first introduced by Kesten and Spitzer [1] and Borodin [2,3] in order to construct new self-similar stochastic processes.Kesten and Spitzer [1] proved that when the random walk and the random scenery belong to the domains of attraction of different stable laws of indices 1 <  ≤ 2 and 0 <  ≤ 2, respectively, then there exists  > 1/2 such that { −  ⌊⌋ ,  ≥ 0} converges weakly as  → ∞ to a continuous -self-similar process with stationary increments,  being related to  and  by  = 1 −  −1 + () −1 .The limiting process can be seen as a mixture of -stable processes, but it is not a stable process.When 0 <  < 1 and for arbitrary , the sequence { −1/  ⌊⌋ ,  ≥ 0} converges weakly, as  → ∞, to a stable process with index  (see Castell et al. [4]).Bolthausen [5] (see also Deligiannidis and Utev [6]) gave a method to solve the case  = 1 and  = 2 and, especially, he proved that when  is a recurrent Z 2random walk, the sequence {( log ) −1/2  ⌊⌋ ,  ≥ 0} satisfies a functional central limit theorem.More recently, the case  one-or two-dimensional random walks and  ∈ (0, 2) was solved in Castell et al. [4]; the authors prove that the sequence { −1/ (log ) 1/−1  ⌊⌋ ,  ≥ 0} converges weakly to a stable process with index .Finally for any arbitrary transient random walk, it can be shown that the sequence { −1/2   ,  ∈ N} is asymptotically normal (see for instance Spitzer [7] page 53).
The problem we investigate in the present paper has already been studied in Lewis [24] in the case that random sceneries 's satisfy E[(0)] = 0 and E[ 2 (0)] = 1, and the random walk  (which can be Z  -valued) satisfies some mild conditions.Lewis [24] established the following LIL: where   () is the number of visits of the random walk to the point  ∈ Z in the time interval [0, ], i.e., Here and in the sequel, the following notation is used: for  > 0 and  ≥ 0, It is therefore natural to investigate limit behavior of RWRS   when the sceneries 's do not have finite second moment.For the sake of convenience, we are summarizing here the main assumptions we are making on the sceneries 's.Assume that the sceneries 's belong to the domain of attraction of a stable law   (0 <  < 2); that is, 's satisfy that lim where   is a stable distribution of index 0 <  < 2, with characteristic function for some From the known characterization of the domain of attraction of a stable law   (Feller [25], II, Chap.17) it follows that, for 0 <  < 2, (5) and ( 6) are equivalent to as  → ∞ for suitable constants  1,1 and  1,2 .Note that ( 5) and ( 6) imply For  = 1 we impose an additional condition (stronger than (5) and ( 6)), namely, that for some positive constant  0 , It is well known that LILs for heavy tailed random variables are different from those for random variables attracted to the normal law.We have to use power norming and the resulting limit theorem is called Chover-type LIL (see Chover [26]).The main results of this paper read as follows.
Theorem 1.Let  = {(),  ∈ Z} be a sequence of i.i.d.random variables satisfying (5) and (9) for all large .Letting  ↓ 0, it yields that the limit superior on left-hand side of (10) is less than  1/ .Equation (12) implies that for infinitely many .Letting  ↓ 0, it yields that the limit superior on left-hand side of ( 10) is greater than  1/ .Moreover, from the proof of Theorem 1 below, the upper bound of (10) does not need the assumptions that  is supported on [0, ∞) and absolutely continuous.
Complementary to Theorem 1 we have the following clustering statement, which gives additional information about the path behavior of RWRS   .Theorem 4.Under the assumptions of Theorem 1, with probability one, every point in the interval (1,  1/ ] is a cluster point of the sequence: Throughout this paper, we use the notations: o. mean infinitely often, a.s.mean almost surely, E[⋅] mean expectation, and E F [⋅] mean conditional expectation given -field F. An unspecified positive and finite constant will be denoted by , which may not be the same in each occurrence.More specific constants in Section  are numbered as  ,1 ,  ,2 , . ... The sign ⌊⋅⌋ sometimes denotes the integer part anf at other times denotes usual brackets; it will be clear from the context.Since we shall deal with index  which ultimately tends to infinity, our statements, sometimes without further mention, are valid only when  is sufficiently large.

Preliminaries
In this section we investigate some technical results necessary for our argumentation.We will first present a version of the Borel-Cantelli lemma to sums of conditional probabilities (see, e.g., Theorem 2.8.5 in Stout [27]).Lemma 5. Let {  ,  ≥ 1} be a sequence of arbitrary events and {G  ,  ≥ 1} be an increasing sequence of -fields such that that is, ∑ ∞ =1 P(  | G −1 ) < ∞ implies that   occur at most finitely often and ∑ ∞ =1 P(  | G −1 ) = ∞ implies that   occur infinitely often.
We will need the following large deviation inequalities for RWRS, which may be of independent interest.Lemma 6.Let {(),  ∈ Z} be a sequence of i.i.d.random variables satisfying (5) and (9), and {  ,  ≥ 1} be a sequence of arbitrary random variables and independent of 's.Let {  ,  ≥ 1} be a sequence of positive numbers such that   → ∞.Then Proof.We denote by F = ( 1 ,  2 , . ..) the -field generated by the random walk and By (7), for all  > 0 and Thus, By (19), for all  ∈ Z, It follows that On the other hand, if  ∈ (0, 1), for all  ∈ Z; if  = 1, by ( 9), for all  ∈ Z; and, if  ∈ (1, 2), by ( 8) and ( 19), for all  ∈ Z.Hence, by ( 23)-( 25) and making use of the fact that we have Thus, by ( 22) and (26), Noting that we can rewrite   as we have that It follows from ( 20), (27), and (29) that By replacing () with −(), we have This, together with (30), yields It yields the right-hand side of (17).
We will also need the following two technical results.
we have For the sake of convenience, we denote and W = ∑ ∈S    () for  > 0,  ∈ Z and  −1 <  ≤   .By ( 19) and (44), following the same argument as the proof of ( 29), we have as  → ∞ for 0 <  < 2. On the other hand, by (22) and noting we have for  −1 <  ≤   and 0 <  < 2, It follows that From Newman and Wright [28], we call a finite collection of random variables   , 1 ≤  ≤ , which is associated if any two coordinatewise nondecreasing functions  1 ,  2 on R  such that   =   ( 1 , . . .,   ) have finite variance for  = 1,2, cov( 1 ,  2 ) ≥ 0; an infinite collection is associated if every finite subcollection is associated.It is not difficult to demonstrate that independent variables are always associated.Moreover, given F,   () − E[  () | F] are nonincreasing functions on  and are also associated variables by Esary et al. [29].Consequently, by Theorem 2 of Newman and Wright [28] and (52), Discrete Dynamics in Nature and Society Hence Note that P ( max Thus, by ( 47), ( 54), (55) and making use of Borel-Cantelli lemma, lim sup Letting  ↓ 0, we obtain (49).The proof of Lemma 8 is completed.