Journal of Probability and StatisticsVolume 2012 (2012), Article ID 371025, 7 pagesdoi:10.1155/2012/371025

Research Article

On Tightness of the Skew Random Walks

Youngsoo Seol

Department of Mathematics, Iowa State University, Ames, IA 50010, USA

Received 8 August 2011; Accepted 5 December 2011

Academic Editor: Nikolaos E. Limnios

Copyright © 2012 Youngsoo Seol. 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.

The primary purpose of this paper is to prove a tightness of -skew random walks. The tightness result implies, in particular, that the -skew Brownian motion can be constructed as the scaling limit of such random walks. Our proof of tightness is based on a fourth-order moment method.

1. Introduction and Statement of the Main Result

Skew Brownian motion was introduced by Itô and Mckean [1] to furnish a construction of certain stochastic processes related to Feller’s classification of second-order differential operators associated with diffusion processes (see also Section  4.2 in [2]). For , the -skew Brownian motion is defined as a one-dimensional Markov process with the same transition mechanism as of the usual Brownian motion, with the only exception that the excursions away from zero are assigned a positive sign with probability and a negative sign with probability . The signs form an i.i.d. sequence and are chosen independently of the past history of the process. If , the process is the usual Brownian motion.

Formally, the -skew random walk on starting at 0 is defined as the birth-death Markov chain with and one-step transition probabilities given by In the special case , is a simple symmetric random walk on . Notice that when , the jumps (in general, increments) of the random walk are not independent.

Harrison and Shepp [3] asserted (without proof) that the functional central limit theorem (FCLT, for short) for reflecting Brownian motion can be used to construct skew Brownian motion as the limiting process of a suitably modified symmetric random walk on the integer lattice. This result has served as a foundation for numerical algorithms tracking moving particle in a highly heterogeneous porous media; see, for instance, [4–7]. In [5] it was suggested that tightness could be obtained based on second moments; however this is not possible even in the case of simple symmetric random walk. The lack of statistical independence of the increments makes a fourth moment proof all the more challenging. Although proofs of FCLTs in more general frameworks have subsequently been obtained by other methods, for example, by Skorokhod embedding in [8], a self-contained simple proof of tightness for simple skew random walk has not been available in the literature.

The main goal of this paper is to prove the following result. Let be the space of continuous functions from into , equipped with the topology of uniform convergence on compact sets. For , let denote the following linear interpolation of : Here and henceforth denotes the integer part of a real number .

Theorem 1.1. For any , there exists a constant , such that the inequality holds uniformly for all , and .

The results stated above implies the following (see, for instance, [9, page 98]).

Corollary 1.2. The family of processes , , is tight in .

2. Proof of Theorem 1.1

In this section we complete the proof of our main result, Theorem 1.1. In what follows we will use to denote the simple symmetric random walk . The following observations can be found in [3].

Proposition 2.1. (a)   has the same distribution as on . That is, is a simple symmetric random walk on , reflected at 0. (b) The processes and have the same distribution.

The next statement describes -step transition probabilities of the skew random walks by relating them to those of (see, for instance, [5, page 436]).

Proposition 2.2. For ,

The following observation is evident from the explicit form of the distribution function of , given in Proposition 2.2.

Proposition 2.3. With probability one,

To show the result of Theorem 1.1, we will need a corollary to Karamata’s Tauberian theorem, which we are going now to state. For a measure on , denote by the Laplace transform of . The transform is well defined for , where is a nonnegative constant, possibly . If and are measures on such that and both exist for all , then the convolution has the Laplace transform for . If is a discrete measure concentrated on , one can identify with a sequence of its values on . For such discrete measures, we have the following. (see, e.g., Corollary  8.10 in [10, page 118]).

Proposition 2.4. Let , , where is a sequence of nonnegative numbers. For slowly varying at infinity and one has if and only if

Here and henceforth, for two sequence of real numbers and means .

We are now in a position to prove the following key proposition. Define a sequence as follows Note that in view of Proposition 2.2,

Proposition 2.5. (a)If then .(b)If then .

Proof. For , let . Notice that is well defined since for . Since , we have Notice that, using the notation of Proposition 2.4, if . Therefore, while . Thus claims (a) and (b) of the proposition follow from Proposition 2.4 applied, respectively, with , for and with , for .

The last technical lemma we need is the following claim.

Lemma 2.6. For integers define Then there is a constant such that

Proof. Using Proposition 2.3, the Markov property, and the fact the excursions of away from zero are the same as excursions of the simple symmetric random walk , we obtain Therefore, Using Proposition 2.5, we obtain for some constant and any . Similarly, Hence, using again Proposition 2.5, for some constant and any .To conclude the proof of the lemma, set .

We are now in a position to complete the proof of our main result.

Completion of the Proof of Theorem 1.1First consider the case where are grid points. Then for a large enough constant .To conclude the proof of Theorem 1.1, it remains to observe that for nongrid points and one can use an approximation by neighbor grid points. In fact, the approximation argument given in [9, pages 100-101] for regular random walks goes through verbatim.

Acknowledgments

The author would like to thank Professor Edward C. Waymire for suggesting this problem and for helpful comments. He also wants to thank Professor Alexander Roitershtein for helpful suggestions and corrections.

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