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Journal of Probability and Statistics
VolumeΒ 2012Β (2012), Article IDΒ 371025, 7 pages
http://dx.doi.org/10.1155/2012/371025
Research Article

On Tightness of the Skew Random Walks

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

Received 8 August 2011; Accepted 5 December 2011

Academic Editor: Nikolaos E.Β Limnios

Copyright Β© 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 𝛼-skew random walks. The tightness result implies, in particular, that the 𝛼-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 ItΓ΄ 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 π›Όβˆˆ(0,1), the 𝛼-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 𝛼 and a negative sign with probability 1βˆ’π›Ό. The signs form an i.i.d. sequence and are chosen independently of the past history of the process. If 𝛼=1/2, the process is the usual Brownian motion.

Formally, the 𝛼-skew random walk on β„€ starting at 0 is defined as the birth-death Markov chain 𝑆(𝛼)={π‘†π‘˜(𝛼);π‘˜β‰₯0} with 𝑆𝛼0=0 and one-step transition probabilities given by 𝑃𝑆(𝛼)π‘˜+1=π‘š+1βˆ£π‘†π‘˜(𝛼)=ξƒ―1=π‘šπ›Όifπ‘š=0,2𝑃𝑆,otherwise,(𝛼)π‘˜+1=π‘šβˆ’1βˆ£π‘†π‘˜(𝛼)=ξƒ―1=π‘š1βˆ’π›Όifπ‘š=0,2,otherwise.(1.1) In the special case 𝛼=1/2, 𝑆(1/2) is a simple symmetric random walk on β„€. Notice that when 𝛼≠1/2, 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 𝐢(ℝ+,ℝ) be the space of continuous functions from ℝ+=[0,∞) into ℝ, equipped with the topology of uniform convergence on compact sets. For π‘›βˆˆβ„•, let 𝑋𝑛(𝛼)∈𝐢(ℝ+,ℝ) denote the following linear interpolation of 𝑆(𝛼)[𝑛𝑑]: 𝑋𝑛(𝛼)1(𝑑)=βˆšπ‘›ξ‚€π‘†[](𝛼)𝑛𝑑[]+(π‘›π‘‘βˆ’π‘›π‘‘)⋅𝑆[](𝛼)𝑛𝑑+1.(1.2) Here and henceforth [π‘₯] denotes the integer part of a real number π‘₯.

Theorem 1.1. For any π›Όβˆˆ(0,1), there exists a constant 𝐢>0, such that the inequality 𝐸||𝑋𝑛(𝛼)(𝑑)βˆ’π‘‹π‘›(𝛼)(||𝑠)4≀𝐢|π‘ βˆ’π‘‘|2,(1.3) holds uniformly for all 𝑠,𝑑>0, and π‘›βˆˆβ„•.

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

Corollary 1.2. The family of processes 𝑋𝑛(𝛼), π‘›βˆˆβ„•, is tight in 𝐢(ℝ+,ℝ).

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 𝑆 to denote the simple symmetric random walk 𝑆(1/2). The following observations can be found in [3].

Proposition 2.1. (a)  |𝑆(𝛼)| has the same distribution as |𝑆| on β„€+={0,1,2,…}. That is, |𝑆(𝛼)| is a simple symmetric random walk on β„€+, reflected at 0.
(b) The processes βˆ’π‘†(𝛼) and 𝑆(1βˆ’π›Ό) have the same distribution.

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

Proposition 2.2. For π‘šβˆˆπ‘, π‘˜>0π‘ƒξ‚€π‘†π‘˜(𝛼)=⎧βŽͺ⎨βŽͺβŽ©ξ€·||𝑆=π‘šπ›Όβ‹…π‘ƒπ‘˜||ξ€Έξ€·||𝑆=π‘šπ‘–π‘“π‘š>0(1βˆ’π›Ό)β‹…π‘ƒπ‘˜||𝑃||𝑆=βˆ’π‘šπ‘–π‘“π‘š<0π‘˜(𝛼)||||𝑆=0=π‘ƒπ‘˜||ξ€Έ=0π‘–π‘“π‘š=0.(2.1)

The following observation is evident from the explicit form of the distribution function of π‘†π‘˜(𝛼), given in Proposition 2.2.

Proposition 2.3. With probability one, 𝐸𝑆(𝛼)𝑗+1βˆ’π‘†π‘—(𝛼)βˆ£π‘†π‘—(𝛼)=(2π›Όβˆ’1)𝟏{𝑆𝑗(𝛼)=0},𝐸𝑆(𝛼)𝑖+1βˆ’π‘†π‘–(𝛼)2βˆ£π‘†π‘–(𝛼)ξ‚Ή=1.(2.2)

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 πœ‡ on [0,∞), denote by βˆ«ξπœ‡(πœ†)∢=∞0π‘’βˆ’πœ†π‘₯πœ‡(𝑑π‘₯) the Laplace transform of πœ‡. The transform is well defined for πœ†βˆˆ(𝑐,∞), where 𝑐>0 is a nonnegative constant, possibly +∞. If πœ‡ and 𝜈 are measures on [0,∞) such that ξπœ‡(πœ†) and Μ‚πœˆ(πœ†) both exist for all πœ†>0, then the convolution 𝛾=πœ‡βˆ—πœˆ has the Laplace transform ̂𝛾(πœ†)=ξπœ‡(πœ†)Μ‚πœˆ(πœ†) for πœ†>0. If πœ‡ is a discrete measure concentrated on β„€+, one can identify πœ‡ with a sequence πœ‡π‘› of its values on π‘›βˆˆβ„€+. For such discrete measures, we have the following. (see, e.g., Corollary  8.10 in [10, page 118]).

Proposition 2.4. Let βˆ‘ξ‚πœ‡(𝑑)=βˆžπ‘›=0πœ‡π‘›π‘‘π‘›, 0≀𝑑<1, where {πœ‡π‘›}βˆžπ‘›=0 is a sequence of nonnegative numbers. For 𝐿 slowly varying at infinity and 0β‰€πœƒ<∞ one has ξ‚πœ‡(𝑑)∼(1βˆ’π‘‘)βˆ’πœƒπΏξ‚€11βˆ’π‘‘π‘Žπ‘ π‘‘β†‘1(2.3) if and only if 𝑛𝑗=0πœ‡π‘—βˆΌ1𝑛Γ(πœƒ)πœƒπΏ(𝑛)π‘Žπ‘ π‘›βŸΆβˆž.(2.4)

Here and henceforth, π‘Žπ‘›βˆΌπ‘π‘› for two sequence of real numbers {π‘Žπ‘›}π‘›βˆˆβ„• and {𝑏𝑛}π‘›βˆˆβ„• means limπ‘›β†’βˆžπ‘Žπ‘›/𝑏𝑛=1.

We are now in a position to prove the following key proposition. Define a sequence {π‘ž(π‘˜)}π‘˜βˆˆβ„€+ as follows ⎧βŽͺ⎨βŽͺβŽ©ξ‚΅π‘–ξ‚Ά2𝑔(π‘˜)=0ifπ‘˜βˆˆβ„•isodd2π‘–βˆ’2𝑖ifπ‘˜=2π‘–βˆˆβ„•iseven.(2.5) Note that in view of Proposition 2.2,𝑆𝑔(π‘˜)=π‘ƒπ‘˜ξ€Έξ€·||𝑆=0=π‘ƒπ‘˜||ξ€Έξ‚€||𝑆=0=π‘ƒπ‘˜(𝛼)||𝑆=0=π‘ƒπ‘˜(𝛼)=0.(2.6)

Proposition 2.5. (a)If πœ‡(𝑗)=π‘”βˆ—π‘”(𝑗) then βˆ‘π‘šπ‘—=0πœ‡(𝑗)βˆΌπ‘š.(b)If 𝜈(𝑗)=π‘”βˆ—π‘”βˆ—π‘”βˆ—π‘”(𝑗) then βˆ‘π‘šπ‘—=0𝜈(𝑗)βˆΌπ‘š2.

Proof. For π‘‘βˆˆ(0,1), let βˆ‘Μƒπ‘”(𝑑)=βˆžπ‘˜=0𝑔(π‘˜)π‘‘π‘˜. Notice that ̃𝑔(𝑑) is well defined since 𝑔(π‘˜)=𝑃(π‘†π‘˜=0)<1 for π‘˜β‰₯0. Since 𝑔(2𝑗)=𝑗2𝑗2βˆ’2𝑗=(βˆ’1)π‘—ξ€·π‘—βˆ’1/2ξ€Έ, we have ̃𝑔(𝑑)=βˆžξ“π‘˜=0𝑔(π‘˜)π‘‘π‘˜=βˆžξ“π‘—=0βŽ›βŽœβŽœβŽπ‘—βŽžβŽŸβŽŸβŽ 22π‘—βˆ’2𝑗𝑑2𝑗=βˆžξ“π‘—=0(βˆ’1)π‘—βŽ›βŽœβŽœβŽœβŽβˆ’12π‘—βŽžβŽŸβŽŸβŽŸβŽ π‘‘2𝑗=βˆžξ“π‘—=0βŽ›βŽœβŽœβŽœβŽβˆ’12π‘—βŽžβŽŸβŽŸβŽŸβŽ ξ€·βˆ’π‘‘2𝑗=ξ€·1βˆ’π‘‘2ξ€Έβˆ’1/2.(2.7) Notice that, using the notation of Proposition 2.4, ̃𝑔(𝑑)=̂𝑔(πœ†) if 𝑑=π‘’βˆ’πœ†. Therefore, ξ‚πœ‡(𝑑)=̃𝑔2(𝑑)=(1βˆ’π‘‘2)βˆ’1 while Μƒπœˆ(𝑑)=̃𝑔4(𝑑)=(1βˆ’π‘‘2)βˆ’2. Thus claims (a) and (b) of the proposition follow from Proposition 2.4 applied, respectively, with πœƒ=1, 𝐿=1 for πœ‡ and with πœƒ=2, 𝐿=1 for 𝜈.

The last technical lemma we need is the following claim.

Lemma 2.6. For integers 0<𝑖1<𝑖2<𝑖3<𝑖4 define 𝐴𝑖1,𝑖2,𝑖3ξ€Έξ‚€π‘†βˆΆ=𝐸𝑖(𝛼)3+1βˆ’π‘†π‘–(𝛼)32𝑆𝑖(𝛼)2+1βˆ’π‘†π‘–(𝛼)2𝑆𝑖(𝛼)1+1βˆ’π‘†π‘–(𝛼)1,𝐡𝑖1,𝑖2,𝑖3,𝑖4ξ€Έξ‚€π‘†βˆΆ=𝐸𝑖(𝛼)4+1βˆ’π‘†π‘–(𝛼)4𝑆𝑖(𝛼)3+1βˆ’π‘†π‘–(𝛼)3𝑆𝑖(𝛼)2+1βˆ’π‘†π‘–(𝛼)2𝑆𝑖(𝛼)1+1βˆ’π‘†(𝛼)i1.(2.8) Then there is a constant 𝐢>0 such that 1≀𝑖1<𝑖2<𝑖3β‰€π‘˜βˆ’π‘—π΄ξ€·π‘–1,𝑖2,𝑖3ξ€Έ||||β‰€πΆπ‘˜βˆ’π‘—2,1≀𝑖1<𝑖2<𝑖3<𝑖4β‰€π‘˜βˆ’π‘—π΅ξ€·π‘–1,𝑖2,𝑖3,𝑖4ξ€Έ||||β‰€πΆπ‘˜βˆ’π‘—2.(2.9)

Proof. Using Proposition 2.3, the Markov property, and the fact the excursions of 𝑆(𝛼) away from zero are the same as excursions of the simple symmetric random walk 𝑆, we obtain 𝐴𝑖1,𝑖2,𝑖3𝑆=𝐸𝑖(𝛼)3+1βˆ’π‘†π‘–(𝛼)32𝑆𝑖(𝛼)2+1βˆ’π‘†π‘–(𝛼)2𝑆𝑖(𝛼)1+1βˆ’π‘†π‘–(𝛼)1ξ‚πŸ{𝑆𝑖1(𝛼)=0}𝟏{𝑆𝑖2(𝛼)=0}𝑆=𝑃𝑖1𝑆=0β‹…(2π›Όβˆ’1)⋅𝑃𝑖2=0βˆ£π‘†π‘–1ξ€Έ=0β‹…(2π›Όβˆ’1)=(2π›Όβˆ’1)2𝑔𝑖1𝑔𝑖2βˆ’π‘–1ξ€Έ.(2.10) Therefore, 1≀𝑖1<𝑖2<𝑖3<β‰€π‘˜βˆ’π‘—π΄ξ€·π‘–1,𝑖2,𝑖3≀[π‘˜βˆ’π‘—]𝑖3𝑖=03βˆ’1𝑖2𝑖=02βˆ’1𝑖1=0𝑔𝑖2βˆ’π‘–1𝑔𝑖1ξ€Έ.(2.11) Using Proposition 2.5, we obtain [π‘˜βˆ’π‘—]𝑖3𝑖=03βˆ’1𝑖2𝑖=02βˆ’1𝑖1=0𝑔𝑖2βˆ’π‘–1𝑔𝑖1ξ€Έ=[π‘˜βˆ’π‘—]𝑖3𝑖=03βˆ’1𝑖2=0ξ€·π‘–π‘”βˆ—π‘”2≀[π‘˜βˆ’π‘—]𝑖3=0[π‘˜βˆ’π‘—]𝑖2=0ξ€·π‘–π‘”βˆ—π‘”2≀𝐢1||||π‘˜βˆ’π‘—2,(2.12) for some constant 𝐢1>0 and any π‘˜,π‘—βˆˆβ„•.
Similarly, 𝐡𝑖1,𝑖2,𝑖3,𝑖4ξ€Έ=(2π›Όβˆ’1)4𝑆⋅𝑃𝑖1ξ€Έβ‹…=03ξ‘π‘Ž=1π‘ƒξ€·π‘†π‘–π‘Ž+1=0βˆ£π‘†π‘–π‘Žξ€Έ=0=(2π›Όβˆ’1)4𝑔𝑖1𝑔𝑖2βˆ’π‘–1𝑔𝑖3βˆ’π‘–2𝑔𝑖4βˆ’π‘–3ξ€Έ.(2.13) Hence, using again Proposition 2.5, 0≀𝑖1<𝑖2<𝑖3<𝑖4𝐡𝑖1,𝑖2,𝑖3,𝑖4≀[π‘˜βˆ’π‘—]𝑖4=0ξ€·π‘–π‘”βˆ—π‘”βˆ—π‘”βˆ—π‘”4≀𝐢2||||π‘˜βˆ’π‘—2,(2.14) for some constant 𝐢2>0 and any π‘˜,π‘—βˆˆβ„•.
To conclude the proof of the lemma, set 𝐢∢=max{𝐢1,𝐢2}.

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

Completion of the Proof of Theorem 1.1
First consider the case where 𝑠=𝑗/𝑛<π‘˜/𝑛=𝑑 are grid points. Then 𝐸|||||𝑆[](𝛼)π‘›π‘‘βˆšπ‘›βˆ’π‘†[](𝛼)π‘›π‘ βˆšπ‘›|||||4=1𝑛2𝐸||π‘†π‘˜(𝛼)βˆ’π‘†π‘—(𝛼)||4=1𝑛2𝐸|||||π‘˜βˆ’1𝑖=𝑗𝑆(𝛼)𝑖+1βˆ’π‘†π‘–(𝛼)|||||4=1𝑛2π‘˜βˆ’1𝑖=𝑗𝐸𝑆(𝛼)𝑖+1βˆ’π‘†π‘–(𝛼)4+1𝑛2𝑖1<𝑖2β‰€π‘˜βˆ’π‘—πΈξ‚€π‘†π‘–(𝛼)1+1βˆ’π‘†π‘–(𝛼)12𝑆𝑖(𝛼)2+1βˆ’π‘†π‘–(𝛼)22+1𝑛2𝑖1<𝑖2<𝑖3β‰€π‘˜βˆ’π‘—πΈξ‚€π‘†π‘–(𝛼)3+1βˆ’π‘†π‘–(𝛼)32𝑆𝑖(𝛼)2+1βˆ’π‘†π‘–(𝛼)2𝑆𝑖(𝛼)1+1βˆ’π‘†π‘–(𝛼)1+1𝑛2𝑖1<𝑖2<𝑖3<𝑖4β‰€π‘˜βˆ’π‘—πΈξƒ©4ξ‘π‘Ž=1𝑆𝑖(𝛼)π‘Ž+1βˆ’π‘†π‘–(𝛼)π‘Žξƒͺ≀1𝑛2π‘˜βˆ’1𝑖=𝑗11+𝑛2βŽ›βŽœβŽœβŽ2βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽ2⎞⎟⎟⎠+1π‘˜βˆ’π‘—π‘˜βˆ’π‘—π‘›2𝐢1||||π‘˜βˆ’π‘—2+1𝑛2𝐢2||||π‘˜βˆ’π‘—2≀𝐢3|π‘‘βˆ’π‘ |2,(2.15) for a large enough constant 𝐢3>0.
To conclude the proof of Theorem 1.1, it remains to observe that for nongrid points 𝑠 and 𝑑 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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