Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 8968947, 9 pages
https://doi.org/10.1155/2018/8968947

Research Article

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

Wensheng Wang¹ and Anwei Zhu²

¹School of Economics, Hangzhou Dianzi University, Hangzhou 310018, China
²College of Science, Hangzhou Normal University, Hangzhou 310036, China

Correspondence should be addressed to Wensheng Wang; wswang@aliyun.com

Received 12 June 2018; Accepted 19 July 2018; Published 1 August 2018

Academic Editor: Francisco R. Villatoro

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.

Abstract.

Let be a sequence of real valued random variables, and . Let be a sequence of real valued random variables which are independent of ’s. Denote by Kesten-Spitzer random walk in random scenery, where means the unique integer satisfying . It is assumed that ’s belong to the domain of attraction of a stable law with index . 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 scenery . The obtained results supplement to some corresponding results in the literature.

1. Introduction

Let be a sequence of real valued random variables, and . Let be a sequence of -valued random variables which are independent of ’s. We refer to as the random walk and as the random scenery. Then the process is defined by

where and means the unique integer satisfying , 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 and , respectively, then there exists such that converges weakly as to a continuous -self-similar process with stationary increments, being related to and by . The limiting process can be seen as a mixture of -stable processes, but it is not a stable process. When and for arbitrary , the sequence 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 and and, especially, he proved that when is a recurrent -random walk, the sequence satisfies a functional central limit theorem. More recently, the case one- or two-dimensional random walks and was solved in Castell et al. [4]; the authors prove that the sequence converges weakly to a stable process with index . Finally for any arbitrary transient random walk, it can be shown that the sequence is asymptotically normal (see for instance Spitzer [7] page 53).

Among others, we can cite strong approximation results [8–10], laws of the iterated logarithm [11–13], limit theorems for correlated sceneries or walks [14–17], large and moderate deviations results [18–22], and ergodic and mixing properties (see the survey [23]).

The problem we investigate in the present paper has already been studied in Lewis [24] in the case that random sceneries ’s satisfy and , and the random walk (which can be -valued) satisfies some mild conditions. Lewis [24] established the following LIL:

where is the number of visits of the random walk to the point in the time interval , i.e.,

Here and in the sequel, the following notation is used: for and ,

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 ; that is, ’s satisfy that

where is a stable distribution of index , 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 , (5) and (6) are equivalent to

as for suitable constants and . Note that (5) and (6) imply

For we impose an additional condition (stronger than (5) and (6)), namely, that for some positive constant ,

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 be a sequence of i.i.d. random variables satisfying (5) and (9), and be a sequence of i.i.d. random variables with a common distribution and independent of ’s. Assume that is supported on , absolutely continuous, and , . Then

Theorem 1 gives the following information about the maximal growth rate of RWRS .

Corollary 2. We have for all , with probability one,

and

Remark 3. It follows from Corollary 2 that the maximal growth rate of is of the order . Equation (10) is equivalent to (11) and (12). In fact, (11) implies that

for all large . Letting , it yields that the limit superior on left-hand side of (10) is less than . Equation (12) implies that

for infinitely many . Letting , it yields that the limit superior on left-hand side of (10) is greater than . Moreover, from the proof of Theorem 1 below, the upper bound of (10) does not need the assumptions that is supported on 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 is a cluster point of the sequence:

Throughout this paper, we use the notations: if , if and if . Let i.o. mean infinitely often, a.s. mean almost surely, mean expectation, and mean conditional expectation given -field . 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 . 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.

2. 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 be a sequence of arbitrary events and be an increasing sequence of -fields such that for each . Then

that is, implies that occur at most finitely often and implies that occur infinitely often.

We will need the following large deviation inequalities for RWRS, which may be of independent interest.

Lemma 6. Let be a sequence of i.i.d. random variables satisfying (5) and (9), and be a sequence of arbitrary random variables and independent of ’s. Let be a sequence of positive numbers such that . Then

Proof. We denote by the -field generated by the random walk and

By (7), for all and ,

Thus,

By (19), for all ,

It follows that

On the other hand, if ,

for all ; if , by (9),

for all ; and, if , by (8) and (19),

for all . 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).
To verify the left-hand side of (17), we denote by and the events and , , respectively. By (19) and some conditional argument, we have

On the other hand, by (28),

This, together with (20) and (32), yields that

Note that

Thus, by (33)-(36),

It yields the left-hand side of (17). The proof of Lemma 6 is completed.

We will also need the following two technical results.

Lemma 7. Let be a sequence of i.i.d. nonnegative random variables with a common distribution . Assume that is absolutely continuous and , . Then, for all ,

Proof. Let , , and be the inverse of . Let be i.i.d. random variables with the distribution of uniform over and . Let be a constant which will be chosen later on and , . By making use of the fact that is a uniform random variable, we have , . On the other hand, , . Thus, since ’s are nonnegative, and are nonincreasing:

By making use of Borel-Cantelli lemma,

To each , there exists an integer such that . Thus, by (40),

Letting , (38) is proved. The proof of Lemma 7 is completed.

Lemma 8. Let be a sequence of i.i.d. random variables satisfying (5) and (9), and be a sequence of arbitrary random variables and independent of ’s. Let be a nondecreasing sequences of positive integers such that , and . Then

Proof. Let and . Since is increasing and , we have that and . Noting

we have

For the sake of convenience, we denote , ,

and for , and .
By (19) and (44),

It follows that

Since

following the same argument as the proof of (29), we have

as for . On the other hand, by (22) and noting

we have for and ,

It follows that

From Newman and Wright [28], we call a finite collection of random variables , , which is associated if any two coordinatewise nondecreasing functions on such that have finite variance for , cov; 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 , 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),

Hence

Note that

Thus, by (47), (54), (55) and making use of Borel-Cantelli lemma,

By replacing with , following the same argument, we have that (56) also holds if is replaced with . It yields

Therefore, by (57),

It follows that

Letting , we obtain (49). The proof of Lemma 8 is completed.

3. Proofs

Proof of Theorem 1. Let and be two arbitrary constants. Let , , and be defined as in Lemma 8 with . By Chover’s law of the iterated logarithm (see Chover [26] and Qi and Cheng [30]) we have a.s. Thus, for any sample point for which it holds, there exists such that for all and , where is the number of integers belonging to , . It follows that, for all ,

By (28) and (60), we have that a.s. and a.s. Hence, to prove (10), by (28), it suffices to prove that, for all ,

and

By (42), (61) holds. Thus, it remains to prove (62). Let . Let be a constant. For , let and . Since is increasing and , we have that ’s are well-defined, and . Noting if and if , we have, for ,

For , we denote , , and . Denote by , , and the events , , and . Let be the -field generated by . Then, . By (19) and (63),

Similar to (36), we have

Similar to (33) and (35), we have and for all , respectively. Thus, by (64) and (65), we have

By choosing small enough such that , and making use of Lemma 5,

By the definitions of and , we have that

and that there exist integers such that and . It follows that

By Lemma 7, we have almost surely for all large . This, together with (60), (68), and (69), yields

Since , by (70), we have . Thus, (67) remains true when is replaced with . Hence, we have

By (68), (70) and following the same argument as the proof of (32),

Thus, by making use of Borel-Cantelli lemma,

Noting

by (71) and (73),

This yields (62). The proof of Theorem 1 is completed.

Proof of Theorem 4. Fix . Let and . It is enough to prove that

To prove (76), it suffices to prove that, for all , with probability one,

and

By Lemma 6,

By making use of Borel-Cantelli lemma, we obtain (77).
It remains to prove (78). For the case , following the same lines as the proof of (62), we have that there exists a subsequence of the subsequence such that (78) holds. For the case , we have . For , let . Following the same lines as the proof of (62), we have, with probability one,

On the other hand, is a subsequence of the subsequence . Thus, we obtain (78) again. The proof of Theorem 4 is completed.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by NSFC (11671115) and NSF of Zhejiang Province (LY14A010025).

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