# On Tightness of the Skew Random Walks

Authors: Youngsoo Seol
Journal: Journal of Probability and Statistics
 (2012)

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## Abstract

The primary purpose of this paper is to prove a tightness ofα-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.


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## Body

## 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 α∈(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 1-α. 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, theα-skew random walk on ℤ starting at 0 is defined as the birth-death Markov chain S(α)={Sk(α);k≥0} with S0α=0 and one-step transition probabilities given by
(1.1)P(Sk+1(α)=m+1∣Sk(α)=m)={αifm=0,12,otherwise,P(Sk+1(α)=m-1∣Sk(α)=m)={1-αifm=0,12,otherwise.
In the special case α=1/2, S(1/2) is a simple symmetric random walk on ℤ. 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. LetC(ℝ+,ℝ) be the space of continuous functions from ℝ+=[0,∞) into ℝ, equipped with the topology of uniform convergence on compact sets. For n∈ℕ, let Xn(α)∈C(ℝ+,ℝ) denote the following linear interpolation of S[nt](α):
(1.2)Xn(α)(t)=1n(S[nt](α)+(nt-[nt])⋅S[nt]+1(α)).
Here and henceforth [x] denotes the integer part of a real number x.Theorem 1.1.
For anyα∈(0,1), there exists a constant C>0, such that the inequality
(1.3)E|Xn(α)(t)-Xn(α)(s)|4≤C|s-t|2,
holds uniformly for all s,t>0, and n∈ℕ.The results stated above implies the following (see, for instance, [9, page 98]).Corollary 1.2.
The family of processesXn(α), n∈ℕ, is tight in C(ℝ+,ℝ).


## 2. Proof of Theorem1.1

In this section we complete the proof of our main result, Theorem1.1. 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 ℤ+={0,1,2,…}. That is, |S(α)| is a simple symmetric random walk on ℤ+, reflected at 0.
(b) The processes-S(α) and S(1-α) have the same distribution.The next statement describesn-step transition probabilities of the skew random walks by relating them to those of S (see, for instance, [5, page 436]).Proposition 2.2.
Form∈Z, k>0(2.1)P(Sk(α)=m)={α⋅P(|Sk|=m)ifm>0(1-α)⋅P(|Sk|=-m)ifm<0P(|Sk(α)|=0)=P(|Sk|=0)ifm=0.The following observation is evident from the explicit form of the distribution function ofSk(α), given in Proposition 2.2.Proposition 2.3.
With probability one,(2.2)E(Sj+1(α)-Sj(α)∣Sj(α))=(2α-1)1{Sj(α)=0},E[(Si+1(α)-Si(α))2∣Si(α)]=1.To show the result of Theorem1.1, we will need a corollary to Karamata’s Tauberian theorem, which we are going now to state. For a measure μ on [0,∞), denote by μ̂(λ)∶=∫0∞e-λxμ(dx) the Laplace transform of μ. The transform is well defined for λ∈(c,∞), where c>0 is a nonnegative constant, possibly +∞. If μ and ν are measures on [0,∞) such that μ̂(λ) and ν̂(λ) both exist for all λ>0, then the convolution γ=μ*ν has the Laplace transform γ̂(λ)=μ̂(λ)ν̂(λ) for λ>0. If μ is a discrete measure concentrated on ℤ+, one can identify μ with a sequence μn of its values on n∈ℤ+. For such discrete measures, we have the following. (see, e.g., Corollary  8.10 in [10, page 118]).Proposition 2.4.
Letμ̃(t)=∑n=0∞μntn, 0≤t<1, where {μn}n=0∞ is a sequence of nonnegative numbers. For L slowly varying at infinity and 0≤θ<∞ one has
(2.3)μ̃(t)~(1-t)-θL(11-t)ast↑1
if and only if
(2.4)∑j=0nμj~1Γ(θ)nθL(n)asn⟶∞.Here and henceforth,an~bn for two sequence of real numbers {an}n∈ℕ and {bn}n∈ℕ means limn→∞an/bn=1.We are now in a position to prove the following key proposition. Define a sequence{q(k)}k∈ℤ+ as follows
(2.5)g(k)={0ifk∈Nisodd(2ii)2-2iifk=2i∈Niseven.
Note that in view of Proposition 2.2,(2.6)g(k)=P(Sk=0)=P(|Sk|=0)=P(|Sk(α)|=0)=P(Sk(α)=0).Proposition 2.5.
(a)
Ifμ(j)=g*g(j) then ∑j=0mμ(j)~m.(b)
Ifν(j)=g*g*g*g(j) then ∑j=0mν(j)~m2.Proof.
Fort∈(0,1), let g̃(t)=∑k=0∞g(k)tk. Notice that g̃(t) is well defined since g(k)=P(Sk=0)<1 for k≥0. Since g(2j)=(2jj)2-2j=(-1)j(-1/2j), we have
(2.7)g̃(t)=∑k=0∞g(k)tk=∑j=0∞(2jj)2-2jt2j=∑j=0∞(-1)j(-12j)t2j=∑j=0∞(-12j)(-t2)j=(1-t2)-1/2.
Notice that, using the notation of Proposition 2.4, g̃(t)=ĝ(λ) if t=e-λ. Therefore, μ̃(t)=g̃2(t)=(1-t2)-1 while ν̃(t)=g̃4(t)=(1-t2)-2. Thus claims (a) and (b) of the proposition follow from Proposition 2.4 applied, respectively, with θ=1, L=1 for μ and with θ=2, L=1 for ν.The last technical lemma we need is the following claim.Lemma 2.6.
For integers0<i1<i2<i3<i4 define
(2.8)A(i1,i2,i3)∶=E(Si3+1(α)-Si3(α))2(Si2+1(α)-Si2(α))(Si1+1(α)-Si1(α)),B(i1,i2,i3,i4)∶=E(Si4+1(α)-Si4(α))(Si3+1(α)-Si3(α))(Si2+1(α)-Si2(α))(Si1+1(α)-Si1(α)).
Then there is a constant C>0 such that
(2.9)∑1≤i1<i2<i3≤k-jA(i1,i2,i3)≤C|k-j|2,∑1≤i1<i2<i3<i4≤k-jB(i1,i2,i3,i4)≤C|k-j|2.Proof.
Using Proposition2.3, 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
(2.10)A(i1,i2,i3)=E(Si3+1(α)-Si3(α))2(Si2+1(α)-Si2(α))(Si1+1(α)-Si1(α))1{Si1(α)=0}1{Si2(α)=0}=P(Si1=0)⋅(2α-1)⋅P(Si2=0∣Si1=0)⋅(2α-1)=(2α-1)2g(i1)g(i2-i1).
Therefore,
(2.11)∑1≤i1<i2<i3<≤k-jA(i1,i2,i3)≤∑i3=0[k-j]∑i2=0i3-1∑i1=0i2-1g(i2-i1)g(i1).
Using Proposition 2.5, we obtain
(2.12)∑i3=0[k-j]∑i2=0i3-1∑i1=0i2-1g(i2-i1)g(i1)=∑i3=0[k-j]∑i2=0i3-1g*g(i2)≤∑i3=0[k-j]∑i2=0[k-j]g*g(i2)≤C1|k-j|2,
for some constant C1>0 and any k,j∈ℕ.
Similarly,(2.13)B(i1,i2,i3,i4)=(2α-1)4⋅P(Si1=0)⋅∏a=13P(Sia+1=0∣Sia=0)=(2α-1)4g(i1)g(i2-i1)g(i3-i2)g(i4-i3).
Hence, using again Proposition 2.5,
(2.14)∑0≤i1<i2<i3<i4B(i1,i2,i3,i4)≤∑i4=0[k-j]g*g*g*g(i4)≤C2|k-j|2,
for some constant C2>0 and any k,j∈ℕ.
To conclude the proof of the lemma, setC∶=max{C1,C2}.We are now in a position to complete the proof of our main result.Completion of the Proof of Theorem1.1
First consider the case wheres=j/n<k/n=t are grid points. Then
(2.15)E|S[nt](α)n-S[ns](α)n|4=1n2E|Sk(α)-Sj(α)|4=1n2E|∑i=jk-1(Si+1(α)-Si(α))|4=1n2∑i=jk-1E(Si+1(α)-Si(α))4+1n2∑i1<i2≤k-jE(Si1+1(α)-Si1(α))2(Si2+1(α)-Si2(α))2+1n2∑i1<i2<i3≤k-jE(Si3+1(α)-Si3(α))2(Si2+1(α)-Si2(α))(Si1+1(α)-Si1(α))+1n2∑i1<i2<i3<i4≤k-jE(∏a=14Sia+1(α)-Sia(α))≤1n2∑i=jk-11+1n2(k-j2)(k-j2)+1n2C1|k-j|2+1n2C2|k-j|2≤C3|t-s|2,
for a large enough constant C3>0.
To conclude the proof of Theorem1.1, it remains to observe that for nongrid points s and t 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.



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*Source: 371025-2012-05-16.xml*