Optimal Bounds for the Variance of Self-Intersection Local Times

Sergey Utev,George Deligiannidis

doi:10.1155/2016/5370627

Sergey Utev, George Deligiannidis

Open Access

https://doi.org/10.1155/2016/5370627

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Abstract

For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.

Highlights

Introduction and Main ResultsLet X, X1, X2, . . . be independent, identically distributed, Zdvalued random variables, Sn = ∑nj=1 Xj, for n ≥ 1.and define the random walk The special case with P(Xi S0 = fl e) 0, =1/(2d), for all e ∈ Zd with |e| = 1, is known as the simple random walk in Let l(n, x) =Zd and will ∑nj=1 1(Sj =be x) denoted by be the local (SRWn)n∈N0 . time of (Sn)n∈N0 at the site x ∈ Zd, and define for a positive integer α the α-fold self-intersection local timeLn = Ln (α) = ∑ l (n, x)α x∈Zd n (1)= ∑ 1 (Si1 = ⋅ ⋅ ⋅ = Siα ) . i1,...,iα =0
As is obvious from the identities Zn = ∑x∈Zd l(n, x)ξx and var(Zn) = var[Ln(2)] var(ξx), limit theorems for (Zn)n usually require asymptotic results for the local times of the random walk (Sn)n. Such asymptotic results are usually obtained from Fourier techniques applied to the characteristic function f(t) = E[exp(it ⋅ X)], under the additional assumption of a Taylor expansion of the form f(t) = 1 − ⟨Σt, t⟩ + o(|t|2), where Σ is a positive definite covariance matrix [3,4,5,6,7], which further requires that E|X|2 < ∞ and EX = 0
For r = 1, for Several results in [3, 7,8,9,10,11,12,13] are obtained as a special case of Corollary 7 and can be extended to dependent variables, for example, a random walk driven by a hidden Markov chain

Summary

Introduction and Main Results

As is obvious from the identities Zn = ∑x∈Zd l(n, x)ξx and var(Zn) = var[Ln(2)] var(ξx), limit theorems for (Zn)n usually require asymptotic results for the local times of the random walk (Sn)n Such asymptotic results are usually obtained from Fourier techniques applied to the characteristic function f(t) = E[exp(it ⋅ X)], under the additional assumption of a Taylor expansion of the form f(t) = 1 − ⟨Σt, t⟩ + o(|t|2), where Σ is a positive definite covariance matrix [3,4,5,6,7], which further requires that E|X|2 < ∞ and EX = 0. In addition we will compare the self-intersection local times of a general d-dimensional random walk with those of the d-dimensional simple symmetric random walk, (SRWn)n∈N0. For any genuinely d-dimensional random walk with finite second moments and zero mean, the asymptotic behaviour of var(Ln(α)) is similar to that of the d-dimensional simple symmetric random walk. − π2 , 6 and κ1 and κ2 are defined in (58) and (63), respectively. if L󸀠(n, α) is the self-intersection local time of atincofuthnecrtiroanndaolsmo swatailskfi,eisn(d6e)p,etnhednenvtaorf(L(S󸀠nn()αn),)w=hovsaerc(hLanr(aαc)t)e(r1is+o(1))

Proofs

Bounds for Identically Distributed Variables

Proofs of Main Results