On Tightness of the Skew Random Walks

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 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, the 𝛼-skew random walk on ℤ starting at 0 is defined as the birth-death Markov chain 𝑆^(𝛼)={𝑆^𝑘(𝛼);𝑘≥0} with 𝑆^𝛼0=0 and one-step transition probabilities given by 𝑃𝑆^(𝛼)𝑘+1=𝑚+1∣𝑆^𝑘(𝛼)=𝑚=𝛼1if𝑚=0,𝑃𝑆2,otherwise,_(𝛼)𝑘+1=𝑚−1∣𝑆^𝑘(𝛼)=𝑚=11−𝛼if𝑚=0,2,otherwise.(1.1) 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. Let 𝐶(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 𝑋^𝑛(𝛼)∈𝐶(ℝ₊,ℝ) denote the following linear interpolation of 𝑆^{(𝛼)[𝑛𝑡]}: 𝑋^𝑛(𝛼)(𝑡)=1√𝑛𝑆^{[](𝛼)𝑛𝑡}+(𝑛𝑡−[𝑛𝑡])⋅𝑆^{[](𝛼)𝑛𝑡+1}.(1.2) Here and henceforth [𝑥] denotes the integer part of a real number 𝑥.

Theorem 1.1. For any 𝛼∈(0,1), there exists a constant 𝐶>0, such that the inequality 𝐸||𝑋^𝑛(𝛼)(𝑡)−𝑋^𝑛(𝛼)(𝑠)||⁴≤𝐶|𝑠−𝑡|²,(1.3) 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 processes 𝑋^𝑛(𝛼), 𝑛∈ℕ, is tight in 𝐶(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 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 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 𝑚∈𝑍, 𝑘>0𝑃𝑆^𝑘(𝛼)=𝑚=⎧⎪⎨⎪⎩𝛼⋅𝑃||𝑆^𝑘(1−𝛼)⋅𝑃||=||𝑆𝑚_𝑘𝑖𝑓𝑚>0𝑃||𝑆||=−𝑚𝑖𝑓𝑚<0_𝑘(𝛼)||=0=𝑃||𝑆_𝑘||=0𝑖𝑓𝑚=0.(2.1)

The following observation is evident from the explicit form of the distribution function of 𝑆^𝑘(𝛼), given in Proposition 2.2.

Proposition 2.3. With probability one, 𝐸𝑆𝐸^(𝛼)𝑗+1−𝑆𝑆^𝑗(𝛼)_(𝛼)𝑖+1∣𝑆^𝑗(𝛼)=(2𝛼−1)𝟏^{{𝑆𝑗(𝛼)=0}},−𝑆^𝑖(𝛼)²∣𝑆^𝑖(𝛼)=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 𝜇 on [0,∞), denote by 𝜇(𝜆)∶=∫^∞0𝑒^−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 𝜇(𝜆) and 𝜈̂(𝜆) both exist for all 𝜆>0, then the convolution 𝛾=𝜇∗𝜈 has the Laplace transform 𝛾̂(𝜆)=𝜇(𝜆)𝜈̂(𝜆) for 𝜆>0. If 𝜇 is a discrete measure concentrated on ℤ₊, one can identify 𝜇 with a sequence 𝜇_𝑛 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. Let 𝜇(𝑡)=∑^∞𝑛=0𝜇_𝑛𝑡^𝑛, 0≤𝑡<1, where {𝜇_𝑛}^∞𝑛=0 is a sequence of nonnegative numbers. For 𝐿 slowly varying at infinity and 0≤𝜃<∞ one has 𝜇(𝑡)∼(1−𝑡)^−𝜃𝐿11−𝑡𝑎𝑠𝑡↑1(2.3) if and only if ^𝑛_𝑗=0𝜇_𝑗∼1Γ(𝜃)𝑛^𝜃𝐿(𝑛)𝑎𝑠𝑛⟶∞.(2.4)

Here and henceforth, 𝑎_𝑛∼𝑏_𝑛 for two sequence of real numbers {𝑎_𝑛}_𝑛∈ℕ and {𝑏_𝑛}_𝑛∈ℕ means lim_𝑛→∞𝑎_𝑛/𝑏_𝑛=1.

We are now in a position to prove the following key proposition. Define a sequence {𝑞(𝑘)}_𝑘∈ℤ+ as follows 𝑔(𝑘)=⎧⎪⎨⎪⎩02𝑖𝑖2_−2𝑖if𝑘∈ℕisoddif𝑘=2𝑖∈ℕiseven.(2.5) Note that in view of Proposition 2.2,𝑔(𝑘)=𝑃𝑆_𝑘=0=𝑃||𝑆_𝑘||=0=𝑃||𝑆^𝑘(𝛼)||=0=𝑃𝑆^𝑘(𝛼)=0.(2.6)

Proposition 2.5. (a)If 𝜇(𝑗)=𝑔∗𝑔(𝑗) then ∑^𝑚𝑗=0𝜇(𝑗)∼𝑚.(b)If 𝜈(𝑗)=𝑔∗𝑔∗𝑔∗𝑔(𝑗) then ∑^𝑚𝑗=0𝜈(𝑗)∼𝑚².

Proof. For 𝑡∈(0,1), let 𝑔̃(𝑡)=∑^∞𝑘=0𝑔(𝑘)𝑡^𝑘. Notice that 𝑔̃(𝑡) is well defined since 𝑔(𝑘)=𝑃(𝑆_𝑘=0)<1 for 𝑘≥0. Since 𝑔(2𝑗)=^𝑗2𝑗2^−2𝑗=(−1)^𝑗^𝑗−1/2, we have 𝑔̃(𝑡)=^∞_𝑘=0𝑔(𝑘)𝑡^𝑘=^∞_𝑗=0⎛⎜⎜⎝2𝑗𝑗⎞⎟⎟⎠2_−2𝑗𝑡^2𝑗=^∞_𝑗=0(−1)^𝑗⎛⎜⎜⎜⎝−1𝑗2⎞⎟⎟⎟⎠𝑡_2𝑗=^∞_𝑗=0⎛⎜⎜⎜⎝−1𝑗2⎞⎟⎟⎟⎠−𝑡₂^𝑗=1−𝑡²^−1/2.(2.7) Notice that, using the notation of Proposition 2.4, 𝑔̃(𝑡)=𝑔̂(𝜆) if 𝑡=𝑒^−𝜆. Therefore, 𝜇(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 integers 0<𝑖₁<𝑖₂<𝑖₃<𝑖₄ define 𝐴𝑖₁,𝑖₂,𝑖₃∶=𝐸𝑆^𝑖(𝛼)₃₊₁−𝑆^𝑖(𝛼)₃𝐵𝑖₁²𝑆^𝑖(𝛼)₂⁺¹−𝑆^𝑖(𝛼)₂𝑆^𝑖(𝛼)₁⁺¹−𝑆^𝑖(𝛼)₁,,𝑖₂,𝑖₃,𝑖₄∶=𝐸𝑆^𝑖(𝛼)₄₊₁−𝑆^𝑖(𝛼)₄𝑆^𝑖(𝛼)₃⁺¹−𝑆^𝑖(𝛼)₃𝑆^𝑖(𝛼)₂⁺¹−𝑆^𝑖(𝛼)₂𝑆^𝑖(𝛼)₁⁺¹−𝑆^(𝛼)_i1.(2.8) Then there is a constant 𝐶>0 such that _{1≤𝑖1<𝑖}_{2<𝑖3≤𝑘−𝑗1≤𝑖1<𝑖2<}_{𝑖3<𝑖4≤𝑘−𝑗}𝐴𝑖₁,𝑖₂,𝑖₃≤𝐶||𝑘−𝑗||²,𝐵𝑖₁,𝑖₂,𝑖₃,𝑖₄≤𝐶||𝑘−𝑗||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=𝐸𝑆^𝑖(𝛼)₃₊₁−𝑆^𝑖(𝛼)₃²𝑆^𝑖(𝛼)₂⁺¹−𝑆^𝑖(𝛼)₂𝑆^𝑖(𝛼)₁⁺¹−𝑆^𝑖(𝛼)₁𝟏^{{𝑆𝑖1(𝛼)=0}}𝟏_{𝑆^𝑖2(𝛼)_=0}==(𝑃2𝛼𝑆−_𝑖11=)₂0⋅(2𝛼−1)⋅𝑃𝑆_𝑖2=0∣𝑆_𝑖1=0⋅(2𝛼−1)𝑔𝑖₁𝑔𝑖₂−𝑖₁.(2.10) Therefore, _{1≤𝑖1<𝑖2}_{<𝑖3<≤𝑘−𝑗}𝐴𝑖₁,𝑖₂,𝑖₃≤^{[𝑘−𝑗]}_𝑖3𝑖=0³⁻¹_𝑖2𝑖=0²⁻¹_𝑖1=0𝑔𝑖₂−𝑖₁𝑔𝑖₁.(2.11) Using Proposition 2.5, we obtain ^{[𝑘−𝑗]}_𝑖3𝑖=0³⁻¹_𝑖2𝑖=0²⁻¹_𝑖1=0𝑔𝑖₂−𝑖₁𝑔𝑖₁=^{[𝑘−𝑗]}_𝑖3𝑖=0³⁻¹_𝑖2=0𝑔∗𝑔𝑖₂≤^{[𝑘−𝑗]}_{𝑖3=0[𝑘−𝑗]𝑖2=0}≤𝐶₁𝑔∗𝑔𝑖₂||𝑘−𝑗||²,(2.12) for some constant 𝐶₁>0 and any 𝑘,𝑗∈ℕ.
Similarly, 𝐵𝑖₁,𝑖₂,𝑖₃,𝑖₄=(2𝛼−1)⁴⋅𝑃𝑆_𝑖1=0⋅³_𝑎=1=(2𝛼−1)₄𝑃𝑆_𝑖𝑎+1=0∣𝑆_𝑖𝑎=0𝑔𝑖₁𝑔𝑖₂−𝑖₁𝑔𝑖₃−𝑖₂𝑔𝑖₄−𝑖₃.(2.13) Hence, using again Proposition 2.5, _0≤𝑖1<_{𝑖2<𝑖3<𝑖4}𝐵𝑖₁,𝑖₂,𝑖₃,𝑖₄≤^{[𝑘−𝑗]}_𝑖₄₌₀𝑔∗𝑔∗𝑔∗𝑔𝑖₄≤𝐶₂||𝑘−𝑗||²,(2.14) for some constant 𝐶₂>0 and any 𝑘,𝑗∈ℕ.
To conclude the proof of the lemma, set 𝐶∶=max{𝐶₁,𝐶₂}.

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<𝑘/𝑛=𝑡 are grid points. Then 𝐸|||||𝑆^{[](𝛼)𝑛𝑡}√𝑛−𝑆√𝑛^{[](𝛼)𝑛𝑠}|||||⁴=1𝑛²𝐸||𝑆^𝑘(𝛼)−𝑆^𝑗(𝛼)||⁴=1𝑛²𝐸|||||^𝑘−1_𝑖=𝑗𝑆^(𝛼)𝑖+1−𝑆^𝑖(𝛼)|||||⁴=1𝑛^2𝑘−1_𝑖=𝑗𝐸𝑆^(𝛼)𝑖+1−𝑆^𝑖(𝛼)⁴+1𝑛²_{𝑖1<𝑖2≤𝑘−𝑗}𝐸𝑆^𝑖(𝛼)₁₊₁−𝑆^𝑖(𝛼)₁²𝑆^𝑖(𝛼)₂⁺¹−𝑆^𝑖(𝛼)₂²+1𝑛²_{𝑖1<𝑖2<𝑖3≤𝑘−𝑗}+1𝐸𝑆^𝑖(𝛼)₃₊₁−𝑆^𝑖(𝛼)₃²𝑆^𝑖(𝛼)₂⁺¹−𝑆^𝑖(𝛼)₂𝑆^𝑖(𝛼)₁⁺¹−𝑆^𝑖(𝛼)₁𝑛²_{𝑖1<𝑖2<𝑖3<𝑖4≤𝑘−𝑗}𝐸⁴_𝑎=1𝑆^𝑖(𝛼)_𝑎+1−𝑆^𝑖(𝛼)_𝑎≤1𝑛^2𝑘−1_𝑖=𝑗1+1𝑛²⎛⎜⎜⎝𝑘−2𝑗⎞⎟⎟⎠⎛⎜⎜⎝𝑘−2𝑗⎞⎟⎟⎠+1𝑛²𝐶₁||𝑘−𝑗||²+1𝑛²𝐶₂||𝑘−𝑗||²≤𝐶₃|𝑡−𝑠|²,(2.15) for a large enough constant 𝐶₃>0.
To conclude the proof of Theorem 1.1, 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.

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.