Journal of Applied Mathematics

Volume 2012 (2012), Article ID 906373, 13 pages

doi:10.1155/2012/906373

Research Article

Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Kyo-Shin Hwang¹ and Wensheng Wang²

¹Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 25 February 2012; Accepted 19 June 2012

Academic Editor: P. G. L. Leach

Copyright © 2012 Kyo-Shin Hwang and Wensheng Wang. 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.

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

1. Introduction

Let be a sequence of independent and identically distributed random vectors, and write and . Let the renewal process of . A continuous time random walk (CTRW) is defined by

In this setting, represents a particle jump, and is the waiting time preceding that jump, so that represents the particle location after jumps and is the time of the th jump. Then is the number of jumps by time , and the CTRW represents the particle location at time , which is a random walk subordinated to a renewal process.

It should be mentioned that the subordination scheme of CTRW processes is going back to Fogedby [1] and that it was expanded by Baule and Friedrich [2] and Magdziarz et al. [3]. It should also be mentioned that the theory of subordination holds for nonhomogeneous CTRW processes, that were introduced in the following works: Metzler et al. [4, 5] and Barkai et al. [6].

The CTRW is useful in physics for modeling anomalous diffusion. Heavy-tailed particle jumps lead to superdiffusion, where a cloud of particles spreads faster than the classical Brownian motion, and heavy-tailed waiting times lead to subdiffusion. CTRW models and the associated fractional diffusion equations are important in applications to physics, hydrology, and finance; see, for example, Berkowitz et al. [7], Metzler and Klafter [8], Scalas [9], and Meerchaert and Scalas [10] for more information. In applications to hydrology, the heavy tailed particle jumps capture the velocity irregularities caused by a heterogeneous porous media, and the waiting times model particle sticking or trapping. In applications to finance, the particle jumps are price changes or log returns, separated by a random waiting time between trades.

If the jumps belong to the domain of attraction of a stable law with index , , and the waiting times belong to the domain of attraction of a stable law with index , , Becker-Kern et al. [11] and Meerschaert and Scheffler [12] showed that as ,

a non-Markovian limit with scaling , where is a stable Lévy motion and is the inverse or hitting time process of a stable subordinator. Densities of the CTRW scaling limit solve a space-time fractional diffusion equation that also involves a fractional time derivative of order ; see Meerschaert and Scheffler [13], Becker-Kern et al. [11], and Meerschaert and Scheffler [12] for complete details. Becker-Kern et al. [14], Meerschaert and Scheffler [15], and Meerschaert et al. [16] discussed the related limit theorems for CTRWs based on two time scales, triangular arrays and dependent jumps, respectively. The aim of the present paper is to investigate the laws of the iterated logarithm for CTRWs. We establish Chover-type laws of the iterated logarithm for CTRWs with jumps and waiting times in the domains of attraction of stable laws.

Throughout this paper we will use to denote an unspecified positive and finite constant which may be different in each occurrence and use “i.o.” to stand for “infinitely often” and “a.s." to stand for “almost surely” and “” to stand for “”. Our main results read as follows.

Theorem 1.1. Let be a sequence of i.i.d. nonnegative random variables with a common distribution , and let , independent of , be a sequence of i.i.d. nonnegative random variables with a common distribution . Assume that , , where is a slowly varying function, and that is absolutely continuous and , . Let be a sequence such that as . Then one has

The following is an immediate consequence of Theorem 1.1.

Corollary 1.2. If the tail distribution of satisfies in Theorem 1.1, then one has

In the course of our arguments we often make statements that are valid only for sufficiently large values of some index. When there is no danger of confusion, we omit explicit mention of this proviso.

2. Chung Type LIL for Stable Summands

In this section we consider a Chung-type law of the iterated logarithm for sums of random variables in the domain of attraction of a stable law, which will take a key role to show Theorem 1.1. When has a symmetric stable distribution function characterized by

. Chover [17] established that

We call (2.2) as Chover's law of the iterated logarithm. Since then, several papers have been devoted to develop Chover's LIL; see, for example, Hedye [18–20], Pakshirajan and Vasudeva [21], Vasudeva [22], Qi and Cheng [23], Scheffler [24], Chen [25], and Peng and Qi [26] for reference. For some reason the obvious corresponding statement for the “lim  inf” result does not seem to have been recorded, and it is the purpose of this section to do so and may be of independent interest.

Theorem 2.1. Let be a sequence of i.i.d. nonnegative random variables with a common distribution , and let . Assume that is absolutely continuous and , , where is a slowly varying function. Then one has

In order to prove Theorem 2.1, we need some lemmas.

Lemma 2.2. Let be a slowly varying function. Then, if , , one has for any given ,

Proof. See Seneta [27].

Lemma 2.3. Let be a sequence of i.i.d. nonnegative random variables with a common distribution and let . Assume that is absolutely continuous and , , where is a slowly varying function. Then one has for some given small

Proof. We will follow the argument of Lemma 2.1 in Darling [28]. Without loss of generality we can assume since each has a probability of of being the largest term, and for since is presumed continuous.

For notational simplicity we will use the tail distribution and denote by the corresponding density, so that . Then, the joint density of , given , is

Thus

Let us put

so that

It follows from Doeblin's theorem that if ,

for with some large . Then, for , we can choose small enough such that since has regularly varying tail distribution, so that

It follows that

Consider the case . By a slight transformation we find that

Putting

we have since and is small. Thus

By (2.9) and making the change of variable to give

which yields the desired result.

The following large deviation result for stable summands is due to Heyde [19].

Lemma 2.4. Let be a sequence of i.i.d. nonnegative random variables with a common tail distribution satisfying , , where is a slowly varying function. Let be a sequence such that as , and let be a sequence with as . Then

Now we can show Theorem 2.1.

Proof of Theorem 2.1. In order to show (2.3), it is enough to show that for all

We first show (2.18). Let , . Put again . Let be the inverse of . Obverse that , , where is a slowly varying function and , so that

by Lemma 2.2. Let be i.i.d. random variables with the distribution of Uniform over , and let . Then, from the fact that is a Uniform random variable, we note that , . From (2.21), nonnegative, and and nonincreasing, it follows that

Hence, the sum of the left hand side of the previously mentioned probability is finite; by the Borel-Cantelli lemma, we get

Thus, by (2.20) we have

Therefore, by the arbitrariness of , (2.18) holds.

We now show (2.19). Let , . For notational simplicity, we introduce the following notations:

By Lemma 2.3, we have

Thus, we get .

Observe again that and , so that

by Lemma 2.2. Thus, we note

which yields easily . Hence, since , we get . Since are independent, by the Borel-Cantelli lemma, we get

By applying Lemma 2.4 and (2.27) and some simple calculation, we have easily that , so that

which, together with (2.30), implies

This yields (2.19). The proof of Theorem 2.1 is now completed.

3. Proof of Theorem 1.1

Proof of Theorem 1.1. We have to show that for all

We first show (3.1). Let , . For notational simplicity, we introduce the following notations:

By (2.18), we have

Put . Let be the inverse of . Recall that , , where is a slowly varying function, so that and

Note that

Thus, by noting increasing,

Hence, by Lemma 2.2,

Thus, by (3.8) and Lemma 2.4, we have

Therefore, . By the Borel-Cantelli lemma, we get .

Observe that

where stands for the complement of . Thus, letting , we have

which implies that

Thus, by (3.5), we have

This yields (3.1) immediately by letting .

We now show (3.2). Let , . To show (3.2), it is enough to prove

Put

By (2.19), we have

Note that

Thus, by noting increasing,

Hence, by Lemma 2.2,

Similarly, by noting , one can have

Thus, by Lemma 2.4, we have

Therefore, . Since the events are independent, by the Borel-Cantelli lemma, we get .

Now, observe that

Therefore, by letting , we get

which implies (3.14). The proof of Theorem 1.1 is now completed.

Remark 3.1. By the proof Theorem 1.1, (1.3) can be modified as follows:

That is to say that the form of (1.3) is no rare and the variables must be cut down additionally by the factors to achieve a finite lim  sup.

Acknowledgments

The authors wish to express their deep gratitude to a referee for his/her valuable comments on an earlier version which improve the quality of this paper. K. S. Hwang is supported by the Korea Research Foundation Grant Funded by Korea Government (MOEHRD) (KRF-2006-353-C00004), and W. Wang is supported by NSFC Grant 11071076.

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