Journal of Probability and Statistics

Journal of Probability and Statistics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 371025 | https://doi.org/10.1155/2012/371025

Youngsoo Seol, "On Tightness of the Skew Random Walks", Journal of Probability and Statistics, vol. 2012, Article ID 371025, 7 pages, 2012. https://doi.org/10.1155/2012/371025

On Tightness of the Skew Random Walks

Academic Editor: Nikolaos E. Limnios
Received08 Aug 2011
Accepted05 Dec 2011
Published16 May 2012

Abstract

The primary purpose of this paper is to prove a tightness of š›¼-skew random walks. The tightness result implies, in particular, that the š›¼-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 ItĆ“ and Mckean [1] to furnish a construction of certain stochastic processes related to Fellerā€™s classification of second-order differential operators associated with diffusion processes (see also Sectionā€‰ā€‰4.2 in [2]). For š›¼āˆˆ(0,1), the š›¼-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 š›¼ and a negative sign with probability 1āˆ’š›¼. The signs form an i.i.d. sequence and are chosen independently of the past history of the process. If š›¼=1/2, the process is the usual Brownian motion.

Formally, the š›¼-skew random walk on ā„¤ starting at 0 is defined as the birth-death Markov chain š‘†(š›¼)={š‘†š‘˜(š›¼);š‘˜ā‰„0} with š‘†š›¼0=0 and one-step transition probabilities given by š‘ƒī‚€š‘†(š›¼)š‘˜+1=š‘š+1āˆ£š‘†š‘˜(š›¼)ī‚=īƒÆ1=š‘šš›¼ifš‘š=0,2š‘ƒī‚€š‘†,otherwise,(š›¼)š‘˜+1=š‘šāˆ’1āˆ£š‘†š‘˜(š›¼)ī‚=īƒÆ1=š‘š1āˆ’š›¼ifš‘š=0,2,otherwise.(1.1) In the special case š›¼=1/2, š‘†(1/2) is a simple symmetric random walk on ā„¤. Notice that when š›¼ā‰ 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, [4ā€“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 š¶(ā„+,ā„) be the space of continuous functions from ā„+=[0,āˆž) into ā„, equipped with the topology of uniform convergence on compact sets. For š‘›āˆˆā„•, let š‘‹š‘›(š›¼)āˆˆš¶(ā„+,ā„) denote the following linear interpolation of š‘†(š›¼)[š‘›š‘”]: š‘‹š‘›(š›¼)1(š‘”)=āˆšš‘›ī‚€š‘†[](š›¼)š‘›š‘”[]+(š‘›š‘”āˆ’š‘›š‘”)ā‹…š‘†[](š›¼)š‘›š‘”+1ī‚.(1.2) Here and henceforth [š‘„] denotes the integer part of a real number š‘„.

Theorem 1.1. For any š›¼āˆˆ(0,1), there exists a constant š¶>0, such that the inequality šø||š‘‹š‘›(š›¼)(š‘”)āˆ’š‘‹š‘›(š›¼)(||š‘ )4ā‰¤š¶|š‘ āˆ’š‘”|2,(1.3) holds uniformly for all š‘ ,š‘”>0, and š‘›āˆˆā„•.

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

Corollary 1.2. The family of processes š‘‹š‘›(š›¼), š‘›āˆˆā„•, is tight in š¶(ā„+,ā„).

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

Proposition 2.1. (a)ā€‰ā€‰|š‘†(š›¼)| has the same distribution as |š‘†| on ā„¤+={0,1,2,ā€¦}. That is, |š‘†(š›¼)| is a simple symmetric random walk on ā„¤+, reflected at 0.
(b) The processes āˆ’š‘†(š›¼) and š‘†(1āˆ’š›¼) have the same distribution.

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

Proposition 2.2. For š‘šāˆˆš‘, š‘˜>0š‘ƒī‚€š‘†š‘˜(š›¼)ī‚=āŽ§āŽŖāŽØāŽŖāŽ©ī€·||š‘†=š‘šš›¼ā‹…š‘ƒš‘˜||ī€øī€·||š‘†=š‘šš‘–š‘“š‘š>0(1āˆ’š›¼)ā‹…š‘ƒš‘˜||ī€øš‘ƒī‚€||š‘†=āˆ’š‘šš‘–š‘“š‘š<0š‘˜(š›¼)||ī‚ī€·||š‘†=0=š‘ƒš‘˜||ī€ø=0š‘–š‘“š‘š=0.(2.1)

The following observation is evident from the explicit form of the distribution function of š‘†š‘˜(š›¼), given in Proposition 2.2.

Proposition 2.3. With probability one, šøī‚€š‘†(š›¼)š‘—+1āˆ’š‘†š‘—(š›¼)āˆ£š‘†š‘—(š›¼)ī‚=(2š›¼āˆ’1)šŸ{š‘†š‘—(š›¼)=0},šøī‚øī‚€š‘†(š›¼)š‘–+1āˆ’š‘†š‘–(š›¼)ī‚2āˆ£š‘†š‘–(š›¼)ī‚¹=1.(2.2)

To show the result of Theorem 1.1, we will need a corollary to Karamataā€™s Tauberian theorem, which we are going now to state. For a measure šœ‡ on [0,āˆž), denote by āˆ«īšœ‡(šœ†)āˆ¶=āˆž0š‘’āˆ’šœ†š‘„šœ‡(š‘‘š‘„) the Laplace transform of šœ‡. The transform is well defined for šœ†āˆˆ(š‘,āˆž), where š‘>0 is a nonnegative constant, possibly +āˆž. If šœ‡ and šœˆ are measures on [0,āˆž) such that īšœ‡(šœ†) and Ģ‚šœˆ(šœ†) both exist for all šœ†>0, then the convolution š›¾=šœ‡āˆ—šœˆ has the Laplace transform Ģ‚š›¾(šœ†)=īšœ‡(šœ†)Ģ‚šœˆ(šœ†) for šœ†>0. If šœ‡ is a discrete measure concentrated on ā„¤+, one can identify šœ‡ with a sequence šœ‡š‘› of its values on š‘›āˆˆā„¤+. For such discrete measures, we have the following. (see, e.g., Corollaryā€‰ā€‰8.10 in [10, page 118]).

Proposition 2.4. Let āˆ‘ī‚šœ‡(š‘”)=āˆžš‘›=0šœ‡š‘›š‘”š‘›, 0ā‰¤š‘”<1, where {šœ‡š‘›}āˆžš‘›=0 is a sequence of nonnegative numbers. For šæ slowly varying at infinity and 0ā‰¤šœƒ<āˆž one has ī‚šœ‡(š‘”)āˆ¼(1āˆ’š‘”)āˆ’šœƒšæī‚€1ī‚1āˆ’š‘”š‘Žš‘ š‘”ā†‘1(2.3) if and only if š‘›ī“š‘—=0šœ‡š‘—āˆ¼1š‘›Ī“(šœƒ)šœƒšæ(š‘›)š‘Žš‘ š‘›āŸ¶āˆž.(2.4)

Here and henceforth, š‘Žš‘›āˆ¼š‘š‘› for two sequence of real numbers {š‘Žš‘›}š‘›āˆˆā„• and {š‘š‘›}š‘›āˆˆā„• means limš‘›ā†’āˆžš‘Žš‘›/š‘š‘›=1.

We are now in a position to prove the following key proposition. Define a sequence {š‘ž(š‘˜)}š‘˜āˆˆā„¤+ as follows āŽ§āŽŖāŽØāŽŖāŽ©ī‚µš‘–ī‚¶2š‘”(š‘˜)=0ifš‘˜āˆˆā„•isodd2š‘–āˆ’2š‘–ifš‘˜=2š‘–āˆˆā„•iseven.(2.5) Note that in view of Proposition 2.2,ī€·š‘†š‘”(š‘˜)=š‘ƒš‘˜ī€øī€·||š‘†=0=š‘ƒš‘˜||ī€øī‚€||š‘†=0=š‘ƒš‘˜(š›¼)||ī‚ī‚€š‘†=0=š‘ƒš‘˜(š›¼)ī‚=0.(2.6)

Proposition 2.5. (a)If šœ‡(š‘—)=š‘”āˆ—š‘”(š‘—) then āˆ‘š‘šš‘—=0šœ‡(š‘—)āˆ¼š‘š.(b)If šœˆ(š‘—)=š‘”āˆ—š‘”āˆ—š‘”āˆ—š‘”(š‘—) then āˆ‘š‘šš‘—=0šœˆ(š‘—)āˆ¼š‘š2.

Proof. For š‘”āˆˆ(0,1), let āˆ‘Ģƒš‘”(š‘”)=āˆžš‘˜=0š‘”(š‘˜)š‘”š‘˜. Notice that Ģƒš‘”(š‘”) is well defined since š‘”(š‘˜)=š‘ƒ(š‘†š‘˜=0)<1 for š‘˜ā‰„0. Since ī€·š‘”(2š‘—)=š‘—2š‘—ī€ø2āˆ’2š‘—=(āˆ’1)š‘—ī€·š‘—āˆ’1/2ī€ø, we have Ģƒš‘”(š‘”)=āˆžī“š‘˜=0š‘”(š‘˜)š‘”š‘˜=āˆžī“š‘—=0āŽ›āŽœāŽœāŽš‘—āŽžāŽŸāŽŸāŽ 22š‘—āˆ’2š‘—š‘”2š‘—=āˆžī“š‘—=0(āˆ’1)š‘—āŽ›āŽœāŽœāŽœāŽāˆ’12š‘—āŽžāŽŸāŽŸāŽŸāŽ š‘”2š‘—=āˆžī“š‘—=0āŽ›āŽœāŽœāŽœāŽāˆ’12š‘—āŽžāŽŸāŽŸāŽŸāŽ ī€·āˆ’š‘”2ī€øš‘—=ī€·1āˆ’š‘”2ī€øāˆ’1/2.(2.7) Notice that, using the notation of Proposition 2.4, Ģƒš‘”(š‘”)=Ģ‚š‘”(šœ†) if š‘”=š‘’āˆ’šœ†. Therefore, ī‚šœ‡(š‘”)=Ģƒš‘”2(š‘”)=(1āˆ’š‘”2)āˆ’1 while Ģƒšœˆ(š‘”)=Ģƒš‘”4(š‘”)=(1āˆ’š‘”2)āˆ’2. Thus claims (a) and (b) of the proposition follow from Proposition 2.4 applied, respectively, with šœƒ=1, šæ=1 for šœ‡ and with šœƒ=2, šæ=1 for šœˆ.

The last technical lemma we need is the following claim.

Lemma 2.6. For integers 0<š‘–1<š‘–2<š‘–3<š‘–4 define š“ī€·š‘–1,š‘–2,š‘–3ī€øī‚€š‘†āˆ¶=šøš‘–(š›¼)3+1āˆ’š‘†š‘–(š›¼)3ī‚2ī‚€š‘†š‘–(š›¼)2+1āˆ’š‘†š‘–(š›¼)2š‘†ī‚ī‚€š‘–(š›¼)1+1āˆ’š‘†š‘–(š›¼)1ī‚,šµī€·š‘–1,š‘–2,š‘–3,š‘–4ī€øī‚€š‘†āˆ¶=šøš‘–(š›¼)4+1āˆ’š‘†š‘–(š›¼)4š‘†ī‚ī‚€š‘–(š›¼)3+1āˆ’š‘†š‘–(š›¼)3š‘†ī‚ī‚€š‘–(š›¼)2+1āˆ’š‘†š‘–(š›¼)2š‘†ī‚ī‚€š‘–(š›¼)1+1āˆ’š‘†(š›¼)i1ī‚.(2.8) Then there is a constant š¶>0 such that ī“1ā‰¤š‘–1<š‘–2<š‘–3ā‰¤š‘˜āˆ’š‘—š“ī€·š‘–1,š‘–2,š‘–3ī€ø||||ā‰¤š¶š‘˜āˆ’š‘—2,ī“1ā‰¤š‘–1<š‘–2<š‘–3<š‘–4ā‰¤š‘˜āˆ’š‘—šµī€·š‘–1,š‘–2,š‘–3,š‘–4ī€ø||||ā‰¤š¶š‘˜āˆ’š‘—2.(2.9)

Proof. Using Proposition 2.3, the Markov property, and the fact the excursions of š‘†(š›¼) away from zero are the same as excursions of the simple symmetric random walk š‘†, we obtain š“ī€·š‘–1,š‘–2,š‘–3ī€øī‚€š‘†=šøš‘–(š›¼)3+1āˆ’š‘†š‘–(š›¼)3ī‚2ī‚€š‘†š‘–(š›¼)2+1āˆ’š‘†š‘–(š›¼)2š‘†ī‚ī‚€š‘–(š›¼)1+1āˆ’š‘†š‘–(š›¼)1ī‚šŸ{š‘†š‘–1(š›¼)=0}šŸ{š‘†š‘–2(š›¼)=0}ī€·š‘†=š‘ƒš‘–1ī€øī€·š‘†=0ā‹…(2š›¼āˆ’1)ā‹…š‘ƒš‘–2=0āˆ£š‘†š‘–1ī€ø=0ā‹…(2š›¼āˆ’1)=(2š›¼āˆ’1)2š‘”ī€·š‘–1ī€øš‘”ī€·š‘–2āˆ’š‘–1ī€ø.(2.10) Therefore, ī“1ā‰¤š‘–1<š‘–2<š‘–3<ā‰¤š‘˜āˆ’š‘—š“ī€·š‘–1,š‘–2,š‘–3ī€øā‰¤[š‘˜āˆ’š‘—]ī“š‘–3š‘–=03āˆ’1ī“š‘–2š‘–=02āˆ’1ī“š‘–1=0š‘”ī€·š‘–2āˆ’š‘–1ī€øš‘”ī€·š‘–1ī€ø.(2.11) Using Proposition 2.5, we obtain [š‘˜āˆ’š‘—]ī“š‘–3š‘–=03āˆ’1ī“š‘–2š‘–=02āˆ’1ī“š‘–1=0š‘”ī€·š‘–2āˆ’š‘–1ī€øš‘”ī€·š‘–1ī€ø=[š‘˜āˆ’š‘—]ī“š‘–3š‘–=03āˆ’1ī“š‘–2=0ī€·š‘–š‘”āˆ—š‘”2ī€øā‰¤[š‘˜āˆ’š‘—]ī“š‘–3=0[š‘˜āˆ’š‘—]ī“š‘–2=0ī€·š‘–š‘”āˆ—š‘”2ī€øā‰¤š¶1||||š‘˜āˆ’š‘—2,(2.12) for some constant š¶1>0 and any š‘˜,š‘—āˆˆā„•.
Similarly, šµī€·š‘–1,š‘–2,š‘–3,š‘–4ī€ø=(2š›¼āˆ’1)4ī€·š‘†ā‹…š‘ƒš‘–1ī€øā‹…=03ī‘š‘Ž=1š‘ƒī€·š‘†š‘–š‘Ž+1=0āˆ£š‘†š‘–š‘Žī€ø=0=(2š›¼āˆ’1)4š‘”ī€·š‘–1ī€øš‘”ī€·š‘–2āˆ’š‘–1ī€øš‘”ī€·š‘–3āˆ’š‘–2ī€øš‘”ī€·š‘–4āˆ’š‘–3ī€ø.(2.13) Hence, using again Proposition 2.5, ī“0ā‰¤š‘–1<š‘–2<š‘–3<š‘–4šµī€·š‘–1,š‘–2,š‘–3,š‘–4ī€øā‰¤[š‘˜āˆ’š‘—]ī“š‘–4=0ī€·š‘–š‘”āˆ—š‘”āˆ—š‘”āˆ—š‘”4ī€øā‰¤š¶2||||š‘˜āˆ’š‘—2,(2.14) for some constant š¶2>0 and any š‘˜,š‘—āˆˆā„•.
To conclude the proof of the lemma, set š¶āˆ¶=max{š¶1,š¶2}.

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 š‘ =š‘—/š‘›<š‘˜/š‘›=š‘” are grid points. Then šø|||||š‘†[](š›¼)š‘›š‘”āˆšš‘›āˆ’š‘†[](š›¼)š‘›š‘ āˆšš‘›|||||4=1š‘›2šø||š‘†š‘˜(š›¼)āˆ’š‘†š‘—(š›¼)||4=1š‘›2šø|||||š‘˜āˆ’1ī“š‘–=š‘—ī‚€š‘†(š›¼)š‘–+1āˆ’š‘†š‘–(š›¼)ī‚|||||4=1š‘›2š‘˜āˆ’1ī“š‘–=š‘—šøī‚€š‘†(š›¼)š‘–+1āˆ’š‘†š‘–(š›¼)ī‚4+1š‘›2ī“š‘–1<š‘–2ā‰¤š‘˜āˆ’š‘—šøī‚€š‘†š‘–(š›¼)1+1āˆ’š‘†š‘–(š›¼)1ī‚2ī‚€š‘†š‘–(š›¼)2+1āˆ’š‘†š‘–(š›¼)2ī‚2+1š‘›2ī“š‘–1<š‘–2<š‘–3ā‰¤š‘˜āˆ’š‘—šøī‚€š‘†š‘–(š›¼)3+1āˆ’š‘†š‘–(š›¼)3ī‚2ī‚€š‘†š‘–(š›¼)2+1āˆ’š‘†š‘–(š›¼)2š‘†ī‚ī‚€š‘–(š›¼)1+1āˆ’š‘†š‘–(š›¼)1ī‚+1š‘›2ī“š‘–1<š‘–2<š‘–3<š‘–4ā‰¤š‘˜āˆ’š‘—šøīƒ©4ī‘š‘Ž=1š‘†š‘–(š›¼)š‘Ž+1āˆ’š‘†š‘–(š›¼)š‘ŽīƒŖā‰¤1š‘›2š‘˜āˆ’1ī“š‘–=š‘—11+š‘›2āŽ›āŽœāŽœāŽ2āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽ2āŽžāŽŸāŽŸāŽ +1š‘˜āˆ’š‘—š‘˜āˆ’š‘—š‘›2š¶1||||š‘˜āˆ’š‘—2+1š‘›2š¶2||||š‘˜āˆ’š‘—2ā‰¤š¶3|š‘”āˆ’š‘ |2,(2.15) for a large enough constant š¶3>0.
To conclude the proof of Theorem 1.1, it remains to observe that for nongrid points š‘  and š‘” 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.

References

  1. K. Itô and H. P. McKean Jr., ā€œBrownian motions on a half line,ā€ Illinois Journal of Mathematics, vol. 7, pp. 181ā€“231, 1963. View at: Google Scholar | Zentralblatt MATH
  2. K. Itô and H. P. McKean Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, Germany, 1974.
  3. J. M. Harrison and L. A. Shepp, ā€œOn skew Brownian motion,ā€ The Annals of Probability, vol. 9, no. 2, pp. 309ā€“313, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. H. Hoteit, R. Mose, A. Younes, F. Lehmann, and Ph. Ackerer, ā€œThree-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods,ā€ Mathematical Geology, vol. 34, no. 4, pp. 435ā€“456, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. A. Lejay, ā€œOn the constructions of the skew Brownian motion,ā€ Probability Surveys, vol. 3, pp. 413ā€“466, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, and B. Wood, ā€œOccupation and local times for skew Brownian motion with applications to dispersion across an interface,ā€ The Annals of Applied Probability, vol. 21, no. 1, pp. 183ā€“214, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. M. J. Saxton and K. Jacobson, ā€œSingle-particle tracking: applications to membrane dynamics,ā€ Annual Review of Biophysics and Biomolecular Structure, vol. 26, pp. 373ā€“399, 1997. View at: Publisher Site | Google Scholar
  8. A. S. Cherny, A. N. Shiryaev, and M. Yor, ā€œLimit behaviour of the “horizontal-vertical” random walk and some extensions of the Donsker-Prokhorov invariance principle,ā€ Theory of Probability and its Applications, vol. 47, no. 3, pp. 498ā€“517, 2002. View at: Publisher Site | Google Scholar
  9. R. N. Bhattacharya and E. C. Waymire, Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1990.
  10. R. Bhattacharya and E. C. Waymire, A Basic Course in Probability Theory, Universitext, Springer, New York, NY, USA, 2007.

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.


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