# Optimal Bounds for the Variance of Self-Intersection Local Times

Authors: George Deligiannidis; Sergey Utev
Journal: International Journal of Stochastic Analysis
 (2016)

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## Abstract

For aZd-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 liminfn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.


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## Body

## 1. Introduction and Main Results

LetX,X1,X2,… be independent, identically distributed, Zd-valued random variables, and define the random walk S0≔0, Sn=∑j=1n‍Xj, for n≥1. The special case with P(Xi=e)=1/(2d), for all e∈Zd with |e|=1, is known as thesimple random walk in Zd and will be denoted by (SRWn)n∈N0.Letl(n,x)=∑j=1n1(Sj=x) be the local time of (Sn)n∈N0 at the site x∈Zd, and define for a positive integer α the α-foldself-intersection local time(1)Ln=Lnα=∑x∈Zdln,xα=∑i1,…,iα=0n1Si1=⋯=Siα. We will denote the corresponding quantities for simple random walk in Zd by LnSRW(α,d) or simply LnSRW(α) when the dimension is clear from the context.LetR+ and R- be, respectively, the semigroup and the group generated by the support of X, (2)R+≔x∈Zd∣PSn=x>0forsomen≥0,R¯≔x∈Zd∣x=y-zforsomex,y∈R+. Following Spitzer [1], we call the random variable X and the random walk it generatesgenuinelyd-dimensional if the group R¯ is d-dimensional.The quantityLn(α) has received considerable attention in the literature due to its relation toself-avoiding walks andrandom walks in random scenery. In particular let therandom scenery{ξx,x∈Zd} be a collection of i.i.d. random variables, independent of (Sn)n, and define the process Z0=0, Zn=∑i=1n‍ξSi. Then (Zn)n is commonly referred to asrandom walk in random scenery and was introduced in Kesten and Spitzer [2], where functional limit theorems were obtained for Znt under appropriate normalization for the case d=1. The case d=2, with Xi centered with nonsingular covariance matrix, was treated in [3] where it was shown that Z[nt]/nlog⁡n converges weakly to Brownian motion. As is obvious from the identities Zn=∑x∈Zd‍l(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 functionf(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 [3–7], which further requires that E|X|2<∞ and EX=0. Similar restrictions are also required for the application of local limit theorems such as in [8, 9].In this paper, motivated by the results of Spitzer [1] for genuinely d-dimensional random walks and the approach of Becker and König [10], we will study the asymptotic behavior of var(Ln(α)) without imposing any moment assumptions on the random walk. The central idea behind our approach is to compare the self-intersection local times Ln(α) of a general d-dimensional walk with those of its symmetrised version. 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. It is well known that, for some positive constants Kα,d, var⁡(LnSRW(α,d))~Kα,dvd,α(n) as n→∞, for (3)v1,αn≔n1+α,v2,αn≔n2log⁡n2α-4,v3,αn≔nlog⁡n,vd,αn≔n,d≥4. Several other cases have been treated in the literature, using a variety of methods.A careful look at the literature reveals that the most difficult case ind=2 is thenear transient recurrent case, where PSn=0~C/n, which corresponds to genuinely 2-dimensional symmetric recurrent random walks, which will be referred to as a critical case. Surprisingly enough, the variance of the self-intersection local times in the critical case is asymptotically the largest.Theorem 1.
LetX,X1,X2,… be independent, identically distributed, and genuinely d-dimensional Zd-valued random variables, for any d≥1. Then, there exist positive constants Cα,X>cα,X>0, depending on α and the distribution of X, such that for all n large enough (4)var⁡Lnα≤cα,Xvar⁡LnSRWα,d≤Cα,Xvd,αn.The result was motivated by [1, 10] and improves related results of Becker and König for d=3 and d=4. Several cases treated in [3, 4, 10–13] can then be obtained as particular cases.Moreover, we also show the surprising converse. More precisely, we show that the right asymptotic behaviour ofvar(Ln) implies that the jumps must have zero mean and finite second moment.Theorem 2.
LetX,X1,X2,…, be independent, identically distributed, and genuinely d-dimensional with d≤3. If (5)liminfn→∞⁡var⁡Lnαvar⁡LnSRWα>0, then E|X|2<∞ and EX=0.As it follows from Theorem3 given below for d=2,3 and from Theorem 5.2.3 in Chen [12] for d=1, if EX=0 and 0<E|X|2<∞, then liminfnvar⁡(Ln(α))/vd,α(n)>0.For any genuinelyd-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. Also, as it follows from our general bounds (see Proposition 4 and Corollary 7) that the asymptotic results for the genuinely d-dimensional random walk can be reproduced by those of the symmetricone-dimensional random walk with appropriately chosen heavy tails, as was indicated by Kesten and Spitzer [2]. The proofs are based on adapting the Tauberian approach developed in [13].Theorem 3.
Letd=1,2,3, and suppose that for t∈Γ≔[-π,π]d one has(6)ft=1-γt+Rt,ford=1,orft=1-Σt,t+Rt,ford=2,3,where Σ is a nonsingular covariance matrix and R(t)=o(t) for d=1 and o(|t|2) for d=2,3 as t→0. Then (7)
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Moreover, ifL′(n,α) is the self-intersection local time of another random walk, independent of (Sn)n, whose characteristic function also satisfies (6), then var(Ln′(α))=var(Ln(α))(1+o(1)).


## 2. Proofs

### 2.1. General Bounds

We first develop a technique to treat random walks with independent but not necessarily identically distributed increments.Proposition 4 (general upper bound).
Assume thatX1,X2,… are independent Zd-valued random variables and let Su,v≔Xu+⋯+Xu+v. Suppose further that for all n∈N and integers a,u,b,v≥0, with a+u≤b and any x∈Zd, one hasAPSa,u±Sb,v=x≤ϕu+v,BPSa,u=0-PSa,u+Sb,v=0≤ψu,v,where ϕ(u) is nonincreasing and ψ(u,v) is nonincreasing in u and is nondecreasing and subadditive in v in the sense that ψ(u,v+w)≤Aψ[ψ(u,v)+ψ(u,w)], for some constant Aψ independent of u, v, and w. Then, for some constant K=cAψ(1+Aψ)α-2 depending only on α(9)var⁡Lnα≤Kn∑i=0n-1ϕi2α-4∑i,j,k=0n-1ϕj∨iϕk∨i+ϕjψi+k,j.Proof of Proposition4.
We first write out the variance as a sum(10)varLnα=α!2∑k1≤⋯≤kα∑l1≤⋯≤lαPSk1=⋯=Skα,Sl1=⋯=Slα-PSk1=⋯=SkαPSl1=⋯=Slα.An important role is played by the manner in which the two sequences are interlaced, since, for example, if kα≤l1 or lα≤k1, the term vanishes by the Markov property.
We will treat the sum over indices withk1≤l1. The sum over the remaining index set with k1>l1 can be treated in a similar fashion and will contribute a constant factor. Therefore, we assume that k1≤l1 and we arrange the two sequences in an ordered sequence of combined length 2α which we denote as (p1,…,p2α); we also define (ϵ1,…,ϵ2α) where ϵi=0 if pi came from k≔{k1,…,kα} and ϵi=1 if pi came from l≔{l1,…,lα}. Finally we define two new sequences m0,m1,…,m2α-1, and δ1,…,δ2α-1, where m0≔p1,mi=pi+1-pi, and δi=ϵi+1-ϵi, for i=1,…,2α-1. Notice that since we assume that k1≤l1, we have p1=k1 and ϵ1=0. Let v(δ)≔∑i=12α-1|δi| denote theinterlacement index. The terms with v=1 vanish, while the terms with v=2 will be considered separately.
Terms with v ≥ 3. We first consider the sum In over the terms with v≥3 for which we drop the negative part and obtain the bound(11)In≔∑k1≤⋯≤kαl1≤⋯≤lαk1≤l1,vδ≥3PSk1=⋯=Skα,Sl1=⋯=Slα=∑x,y∈Zd∑p1≤⋯≤p2α≤n∑ϵ:vδ≥3PSp1=x,Sp2=x+ϵ2y,…,Sp2α=x+ϵ2αy≤∑x,y∈Zd∑m0,…,m2α-1≤n∑δ:vδ≥3PSm0=xPSm0,m1=δ1y⋯PSm2α-2,m2α-1=δ2α-1y=∑y∈Zd∑m0,…,m2α-1≤n∑δ:vδ≥3PSm0,m1=δ1y⋯PSm2α-2,m2α-1=δ2α-1y.Summing over the free index m0, it is clear that (12)In≤n+1∑m1,…,m2α-1∑y∈Zd∑δ:vδ≥3∏t=12α-1supw⁡PSw,mt=δty.For any δ=(δ1,…,δ2α-1) with v(δ)=v, exactly u≔2α-1-v elements are equal to 0, and therefore by Assumption A with x=0 we have (13)In≤Cn+1∑v=3α∑i=0nϕi2α-1-v∑j1,…,jv=0n∑y∈Zd∑δ′∈-1,+1v∏t=1vsupwt⁡PSwt,jt=δty.Letting (S~n)n∈N0 denote an independent copy of the random walk (Sn)n∈N0 and assuming without loss of generality that j1≤⋯≤jv, we have that for any δ∈{-1,+1}v(14)∑y∈Zd∏t=1vsupwt⁡PSwt,jt=δty≤∏t=2v-1supysupwt⁡PSwt,jt=ysupw1,wv⁡PSw1,j1-δvS~wv,jv=0≤∏t=2v-1ϕjtϕj1+jv≤∏t=2vϕjt∨j1. Let Gn≔∑i=0n‍ϕ(i). Since ϕ is nonincreasing we have that (15)Δn,v≔∑0≤j1≤⋯≤jv≤n∏t=2vϕjt∨j1≤∑jv=0nϕjv∑0≤j1≤⋯≤jv-1≤n∏t=2v-1ϕjt∨j1=GnΔn,v-1,and iterating this procedure, for v≥3, we have that Δn,v≤Δn,3Gnv-3. Combining the two bounds and summing over v=3,…,2α-1, we have that (16)In≤∑v=32α-1cαnGn2α-1-vΔn,v≤cαnGn2α-1-v+v-3Δn,3=cαnGn2α-4Δn,3,where c(α) is a constant depending only on α.
Terms with v = 2. Next we consider the sum Jn over the terms with v=2, which occurs when, for some j, the indices l1,…,lα all lie in [kj,kj+1]. Then it is easy to see that this sum Jn is bounded above by (17)Jn≤Cnsupw0,…,w2α-1∑mα+1,…,m2α-2=0n∏r=α+12α-2PSwr,mr=0∑m0,…,mα=0n∏t=1α-1PSwt,mt=0PSw0,m0+Swα,mα=0-PSw0,m0+⋯+Swα,mα=0≤CnGnα-2supw0,…,wα⁡∑m0,…,mα=0n∏t=1α-1PSwt,mt=0PSw0,m0+Swα,mα=0-PSw0,m0+⋯+Swα,mα=0≤CnGnα-2∑m0,…,mα=0n∏t=1α-1ϕmtψm0+mα,m1+⋯+mα-1≤CαnGnα-2Aψ1+Aψα-2∑m2,…,mα-1∏t=2α-1ϕmt∑m0,m1,mαϕm1ψm0+mα,m1≤CαAψ1+Aψα-2nGn2α-4∑i,j,k=0nϕjψi+k,j.The following corollary provides explicit bounds in the cases that are usually considered in the literature.Corollary 5.
Assume that the conditions of Proposition4 are satisfied with ϕ(m)=Tm-r and ψ(m,k)=Tm-r-1(k∧m). Then,(18)
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.It is straightforward to see that Corollary5 includes random walks with mean zero and finite second moment; for example, d=2 corresponds to r=1 and d=3 to r=3/2. Therefore several relevant results in [3, 7–13] are obtained as a special case of Corollary 5 and extended to the case of independent but not necessarily identically distributed variables, for example, by applying the local limit theorem, as conducted in [8].Also when the random walk incrementX is in the domain of attraction of the one-dimensional symmetric Cauchy law [13, 14] or in the case of planar random walk with second moments [3, 7–9, 11], it is well known that the conditions of Proposition 4 are satisfied with ϕ(m)=T/m and ψ(m,k)=Tm-2(k∧m).However, we can do better for symmetric variables and show that conditionA implies B, which together with the comparison technique motivates the following results. For a real number x, we write [x] for the integer part of x.Proposition 6 (bounds via comparison with characteristic function of symmetric random variables).
LetX1,X2,…, be independent Zd-valued random variables and let fit≔Eexp⁡(itXi). Assume that there exist a measurable function f:Γ→[0,1] and a positive nonincreasing sequence (ϕ(m))m∈N0, such that(19)1-fit≤Tft,fi±t≤ft,∫Γftmdt≤ϕm,for all integers i,m≥0, all t∈Γ, and some positive constant T. Then there exists another positive constant K=c(α,d,T) such that (20)varLnα≤Kn∑i=0n-1ϕi22α-4∑j=0njϕj2∑k=j2nϕk2.Proof of Proposition6.
Using the notation of Proposition4, for positive integers a,u,b,andv, with a+u≤b, ϵj=±1, and any x∈Zd(21)PSa,u+ϵ·Sb,v=x≤12πd∫Γ∏j∈a,a+u∪b,b+vfjϵjtdt≤12πd∫Γftu+vdt≤12πdϕu+v.
To findψ(u,v), notice that since f(t)≥0, (22)ϕm≥∫Γftm1-ftmdt=∑j=0m-1∫Γftm+j1-ftdt≥m∫Γft2m1-ftdt≕mQ2mwhence Q(m)≤2ϕ([m/2])/m. Therefore, (23)PSa,u=0-PSa,u+Sb,1=0=12πd∫Γ∏j=aa+ufjt1-fb+1tdt≤CT∫Γftu1-ftdt≤CTϕu/2u. A telescoping argument implies that (24)PSa,u=0-PSa,u+Sb,v=0≤CTϕu2vu. On the other hand for u≤v we can obtain a tighter bound through(25)PSa,u=0-PSa,u+Sb,v=0≤PSa,u=0≤ϕu. Combining the two bounds above it follows that B is satisfied with ψ(u,v)≔ϕu/2min⁡(u,v)/u. Thus all conditions of Proposition 4 are satisfied and the result follows.The following corollary allows for the case whereϕ(m) is regularly varying.Corollary 7.
Assume that the conditions of Proposition6 are satisfied with ϕ(m)=h(m)m-r, r≥1, where h(·) is slowly varying at ∞. Then,(26)
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.Several results in [3, 7–13] 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. In addition, following [2], we can construct a one-dimensional symmetric random walk with characteristic function f(t)=1-c|t|1/r+o(|t|1/r), where r=2/d for d=2,3 and r=1/2 for d≥4, whose asymptotic behaviour is similar to that of genuinely d-dimensional random walk.The following example of genuinely2-dimensional recurrent walk with infinite variance was motivated by Spitzer [1, pp. 87].Example 8.
LetX1,X2,… be independent, identically distributed, Z2-valued random variables, such that P(|X1|=k)=c/(k3log⁡(k)g), for k≥4 and g∈[0,1). Let (Sn)n∈N0 be the corresponding random walk in Z2. Then we have(27)var⁡Lnα≤cn2max⁡log⁡ng,log⁡log⁡n2α-4log⁡n-21-g,for n≥10. Under these assumptions we have that PSn=0≤c/nlog⁡(n)1-g, which is in thecritical range, where the random walk is recurrent, without second moment. To see why, we note that by a lengthy but straightforward calculation it can be shown that the characteristic function of X satisfies (19) with (28)ϕn=cnlog⁡e∨n1-g,ft=exp⁡-At2ht2,wherehr≔1+log⁡1r+1-g.The sequence ϕ(m) is identified via Fourier inversion, polar coordinates, and a Laplace argument, (29)∫Γftndt≤c∫01exp⁡-nr1+log⁡1r1-g+Oe-n≤cnlog⁡e∨n1-g≕ϕn.


### 2.2. Bounds for Identically Distributed Variables

Proposition 9 (general upper bound for i.i.d.).
LetX,X1,X2,…, be independent, identically distributed, Zd-valued random variables. Suppose that for any x∈Zd and all positive integers a, u, b, and v, with a+u≤b, it holds that(30)PSa,u±Sb,v=x≤ϕu+v,where {ϕ(m)}m∈N0 is a nonincreasing sequence. Then for some constant K=c(α) we have that (31)var⁡Lnα≤Kn∑i=0n-1ϕi2α-4∑j=0njϕj∑k=jαn+1ϕkα.Proof of Proposition9.
By inspecting the proof of Proposition6, we notice that we only need to bound the term Jn. Consider typical ordering (32)0≤i1≤⋯≤ik≤j1≤⋯≤jα≤ik+1≤⋯≤iα≤n, and let us change variables to (m0,…,m2α) such that m0+⋯+m2α=n. Then the contribution to Jn is given by(33)∑m0,…,m2α∏j≠k,k+α1≤j≤2α-1PSmj=0PSmk+mk+α=0-PSmk+⋯+mk+α=0.We keep mj fixed for j≠α,k+α and we sum over m=mk+mk+α from 0 to some M=M(n,{mj}j≠k,k+α). Then for given mk+1,…,mk+α-1, the term in the sum is (34)∑m=0Mm+1PSm=0-PSm+q=0,where q≔mk+1+⋯+mk+α-1. Then since M≤n-q, it is an easy exercise to show that this sum is bounded above by (35)∑m=0Mm+1PSm=0-PSm+q=0≤∑m=0q-1m+1PSm=0+q1n-q≥q∑m=qn-qPSm=0≤∑m=0αm∗∧nm+1PSm=0+αm∗∑m=m∗nPSm=0,where m∗=max⁡mk+1,…,mk+α-1. The result follows by summing over all indices apart from m∗ and changing the order of summation.


### 2.3. Proofs of Main Results

Proof of Theorem1.
We apply a comparison argument found to be useful in many areas (e.g., Montgomery-Smith and Pruss [15], and Lefèvre and Utev [16]). More specifically we bound the quantity var⁡(Ln) by the corresponding quantity for the symmetrised random walk.
Following Spitzer’s argument we notice that withf(t)=E[exp⁡(it·X1)](36)PSa,u+ϵSb,v=x≤c∫Γftuf-tvdt=c∫Γft2u/2f-t2v/2dt.Since |f(t)|2 is the characteristic function of a symmetric random variable in Zd, for some positive λ, we have 1-|f(t)|2≥λ|t|2, and, hence, (37)PSa,u+ϵSb,v=x≤c∫Γexp⁡-λu+v2t2dt≤cu+v-d/2. The result follows from Proposition 9 applied with ϕ(m)=m-d/2.The proof of Theorem2 will be based on the following lemma.Lemma 10.
AssumeX,X1,X2,… are independent, identically distributed, genuinely d-dimensional random variables such that E|X|2=∞. Then there exists a monotone, slowly varying sequence (hn)n∈N0, such that hn→0 as n→∞ and(38)supx∈Zd⁡PSn=x≤cd∫ΓEeit·Xndt≤hnn-d/2.Proof of Lemma10.
Without loss of generality we assume thatX is symmetric. Let σe,L≔E[(e·X)21(|X|≤L)]. Following Spitzer, since X is genuinely d-dimensional, we may assume that there exist positive constants c,W, such that for any unit vector |e|=1 we have that σe,W≥c and 1-f(t)≥c|t|2 for all t∈Γ. Let λd be the d-dimensional Lebesgue measure on Rd and μd the Lebesgue-Haar measure on Sd-1≔{e∈Γ:|e|=1}. Notice that since E|X|2=∞, for any K, we have μd{e:σe,∞<K}=0.
Fix a small positivex such that c/x≥2W, and for any ϵ>0 let K=K(ϵ)=ϵ-d/2. Then there exists L=L(ϵ)>0 small enough so that μd{e:σe,L<K}≤ϵd/2. We partition Sd-1 in two sets (39)AL,K=e∈Sd:σe,L≥K,A¯L,K=e∈Sd:σe,L<K, so that, for any direction e∈A¯L,K, (40)z∈R:1-fze≤x⊆z:cz2≤x⊆z:z≤xc.Hence, using d-dimensional spherical coordinates, (41)λdz,e∈R×A¯L,K:1-fez≤x≤μdA¯L,Kxcd/21d≤ϵd/2xcd/21d. On the other hand, for any t, (42)1-ft=2∑k∈Zdsin⁡t·k22PX=k≥14Et·X2It·X≤12=t24σt/t,1/2t. Now, assume that c/x≥2L. Then for any direction e∈AL,K, by choice of x and since σe,L is increasing in L, for cz2≤1-f(ez)≤x or z≤x/c, it must be the case that (43)x≥1-fez≥z24σe,1/2z≥z24σe,L≥z24K,implying that, on the set AL,K, it must be that |z|≤2x/K. Changing to d-dimensional polar coordinates, we find that (44)λdz,e∈R×AL,K:1-fez≤x≤∫AL,K∫04x/Krd-1drde≤Cdϵd/2xd/2. Overall, for x≤c/4L2, λd{t:1-f(t)≤x}≤cd(xϵ)d/2, and hence {t∈Γ:1-f(t)≤x} has Lebesgue measure o(xd/2).
LetF(x) be the cumulative distribution function of the random variable log⁡(1/f(·)) defined on the probability space Γ with normalised Lebesgue measure. Then F is continuous at x=0 and supported on R+. Moreover, we have that F(x)=o(xd/2) as x↓0. Therefore, for some positive sequence (ϵn)n∈N0 with ϵn→0, we have that(45)12π2∫Γftndt=∫0∞e-nxdFx=n∫0∞e-nxFxdx≤n-d/2ϵn.
It remains to show that there exists a positive, monotone, slowly varying sequence(hn)n∈N0, such that ϵn≤h(n)→0 as n→∞. Let δn=supj≥nϵj and a0≔0 and for n≥1 define an recursively by an=min⁡(2a2r-1,1/δn), for 2r-1<n≤2r, so that an→∞ is monotone, a2r≤2a2r-1 implying that a2n≤4an, and 1/an≥δn≥ϵn. Finally, take hn≔1/max⁡(a0,log⁡an).Proof of Theorem2.
Assume thatE|X|2=∞ and d=2 or d=3. Then, by Lemma 10 there exists a slowly varying sequence hn→0 as n→∞ such that ∫Γ‍|Eexp⁡(it·X)|ndt≤hnn-d/2. Applying Corollary 7 with r=1 and r=3/2 we, respectively, find that (46)
var
⁡
L
n
α
≤
K
n
2
∑
k
=
1
n
h
k
k
2
α
-
4
=
o
n
2
log
⁡
n
2
α
-
4
,
for
d
=
2
,
K
n
∑
k
=
1
n
h
k
2
k
=
o
n
ln
⁡
n
,
for
d
=
3
.
Finally assume thatE|X|2<∞ and E[X]=μ≠0. Then P(Sn=0)=P(Sn′=-nμ) whence it follows that P(Sn=0)=o(n-d/2) (see, e.g., [17, Theorem 2.3.10]). Then inspecting the proof of Proposition 4, one can readily obtain the desired bound for the Jn term, while with slight modification the bound for the In term also follows.
Note that ford=1 the situation is much simpler since then var⁡(LnSRW(α))~C[ELnSRW(α,d)]2 and if E|X|2=∞ or E[X]≠0, ELnSRW(α,d)=o(n(1+α)/2).Proof of Theorem3.
We first give the proof for the cased=1. As in the proof of Proposition 4 we begin from expression (10) and define the sequences pi and δi for i=1,…,2α-1, and the quantity v(δ)=∑i=12α-1‍|δi|. Recall that v(δ) measures the interlacement of the two sequences k1,…,kα and l1,…,lα. For example, v(δ)=1 occurs when either kα≤l1 or lα≤k1, in which case the contribution vanishes by the Markov property. On the other hand v(δ)=2 when, for example, l1,…,lα∈[ki,ki+1] for some i. Finally v(δ)=3 occurs when, for example, (47)k1≤⋯≤kr≤l1≤⋯≤ls≤kr+1≤⋯≤kα≤ls+1≤⋯≤lα≤n.From the proof of Proposition 4, and using the bound P(Sn=0)≤c/n, the terms of the sum are bounded above by n2log(n)2α-1-v(δ), and thus the leading term appears when either v(δ)=2,3, with other terms giving strictly lower order. We will therefore analyze these two situations in detail in order to derive the exact asymptotic constants. When v=3, the two terms in the difference individually give the correct order and will be treated by the classical Tauberian theory. However for v=2, the two terms only give the correct order when considered together. This however forbids the use of Karamata’s Tauberian theorem since the monotonicity restriction would require roughly that Xi is symmetric. Thus the complex Tauberian approach, as developed in [13], is required to justify the answer.
Case  1 ( v ( δ ) = 3 ). Assume that part of the sequence l={l1,…,lα} lies between kr and kr+1 and the rest between ks and ks+1. Then using the change of variables (48)i1=m0,i2=m0+m1,⋮ir=m0+⋯+mr-1j1=m0+⋯+mr,j2=m0+⋯+mr+1,⋮js=m0+⋯+mr+s-1,ir+1=m0+⋯+mr+s,ir+2=m0+⋯+mr+s+1,⋮iα=m0+⋯+mα+s-1,js+1=m0+⋯+mα+s,js+2=m0+⋯+mα+s+1,⋮jα=m2α-1,n=m0+⋯+m2α,we rewrite the positive term in (10) as (49)an=∑PSi1=⋯=Siα;Sj1=⋯=Sjα=∑m0,…,m2α-1∏j=1j≠r,r+s,α+s2α-1PSmj=0PSmr+Smr+s′=Smr+s′+Smα+s′′=0. Notice that from [13] we have that ∑n≥0‍λnP(Sn=0)~log⁡(1/(1-λ))/πγ. Let (50)aλ=1-λ-3-log⁡1-λ2α-4,cγ=πγ-2α+4.Then, by direct calculations and Fourier inversion formula (51)∑n≥0λnan=cγ1-λaλ∑x∈Z∑k1,k2,k3≥0λk1+k2+k3PSk1=xPSk2=-xPSk3=x=cγ1-λaλ12π2∬Γdtds1-λft1-λfs1-λft+s~cγ1-λaλ12π2γ211-λ∬R2dxdy1+x1+y1+x+y~14γ2cγaλ. Next we consider the negative term in (10) (52)bn≔∑m0,…,m2α-1PSm1=⋯=Smr-1=Smr+⋯+Smr+s=Smr+s+1=⋯=Smα+s-1=0PSmr+1=⋯=Smr+s+⋯+Smα+s=Smα+s+1=⋯=Sm2α-1=0. By direct calculations and (6),(53)∑nλnbn=1πγlog⁡11-λα-s+r-21-λ-2∑mr,…,mα+s=0∞λmr+⋯+mα+s∏t=r+1,…,α+s-1t≠r+sPSmt=0PSmr+⋯+Smr+s=0PSmr+s+⋯+Smα+s=0, and using Fourier inversion and (6) the internal sum behaves as (54)2π-α-s+r∫-ππ⋯∫-ππ1-λϕx-11-λϕxϕy-11-λϕy-1∏j=r+1r+s-1∏k=r+s+1α+s-11-λϕxϕtj-11-λϕyϕtk-1dtjdtkdxdy~πγ-α-s+r1-λ-1log⁡11-λα-r+s-2π26.Then, we have ∑n‍λnb(n)~(π2/6(πγ)2α-2)a(λ), whence the Tauberian theorem implies that a(n)-b(n)~n2log(n)2α-4/24π2α-4γ2α-2. Most importantly we see that the lengths and locations of the chains, r and s, do not affect the asymptotic behaviour. Noting that if 1≤r, s≤α-1, we can partition 2α=r+s+(α-r)+(α-s) in (α-1)2 ways, and thus overall the total contribution from terms with v=3 is (55)α!α-1212π2α-4γ2α-2n2logn2α-4.
Case  2 ( v ( δ ) = 2 ). The typical term c(n) was introduced in (33) in the proof of Proposition 9. Now we let λ∈C, with |λ|<1. By lengthy but direct calculations we can derive an expression of the form (56)∑nλncn=α-1γπ2α-2aλ+oaλ,λ⟶1.The approach developed in [13] can then be used to bound the error terms and show that c(n)~[(α-1)/2(γπ)2α-2]n2log(n)2α-4.
Finally taking into account the fact thatl1,…,lα can be in any of the α-1 intervals [ki,ki+1], for i=1,…,α-1, the result follows the overall contribution of terms with v(δ)=2(57)α-122γπ2α-2n2log⁡n2α-4.
The case ford=2 is very similar, so we move on to the case d=3.
Case  3 ( d = 3 and α = 2 ). Using the same notation as before, we have three terms to consider a(n), b(n), and c(n). We first consider c(n). Letting K≔ϵ/1-λ and using the usual power series construction and spherical coordinates(58)∑nλncn=1-λ-22π-6∬J3×J3λfy1-fxdxdy1-λfx21-λfy1-λfxfy~22π-4Σ-11-λ-2∬0Kr4s2drds1+r221+s21+r2+s2~22π-4Σ-1π21-λ-2log⁡11-λ≕κ11-λ-2log11-λ,and thus cn~κ1nlog⁡n, where κ1>0, where the answer can be justified following [13].
The terma(n)-b(n) is trickier to compute. As usual we consider the power series (59)∑n≥0λnan-bn=1-λ-22π-6∬Bϵdxdy1-λfx1-λfy1-λfx+y-1-λ-22π-6∬Bϵdxdy1-λfx1-λfy1-λfxfy=1-λ-22π-6I1λ-I2λ. Let A∈[-1,1] be the cosine of the angle between x and y, which in spherical coordinates is (60)A=Aθ1,θ2,ϕ1,ϕ2=cos⁡ϕ1-ϕ2sin⁡θ1sin⁡θ2+cos⁡θ1cos⁡θ2.Then as 0<λ↑1, using the expansion (6) (61)I1λ~Σ-1∫r,s=0ϵ∫ϕ1,2=02π∫θ1,θ2=0πr2s2sin⁡θ1sin⁡θ2dθ1dθ2dϕ1dϕ2drds1-λ+λr21-λ+λs21-λ+λr2+s2+2Ars=Σ-1∫θ1,θ2=0π∫ϕ1,ϕ2=02π∫r,s=0Ksin⁡θ1sin⁡θ2r2s2dsdrdϕ1dϕ2dθ1dθ21+r21+s21+r2+s2+2Ars~Σ-1log⁡K∫θ1,θ2=0π∫ϕ1,ϕ2=02πsinθ1sinθ2arccos⁡Aθ1,θ2,ϕ1,ϕ21-Aθ1,θ2,ϕ1,ϕ22dϕ1dϕ2dθ1dθ2.The other integral is slightly easier (62)I2λ~Σ-1π2log⁡K∫θ1,θ2=0π∫ϕ1,ϕ2=02πsin⁡θ1sin⁡θ2dϕ1dϕ2dθ1dθ2,and thus overall we must have that(63)I1-I2λ~122π-6Σ-11-λ-2log⁡11-λ∫θ1,θ2=0π∫ϕ1,ϕ2=02πarccos⁡A1-A2-π2sin⁡θ1sin⁡θ2dϕ1dϕ2dθ1dθ2≕κ21-λ-2log⁡11-λ,whence it follows that var⁡(Ln(2))~κ1+κ2nlog⁡n.
To prove the last claim letSn′=X1′+⋯+Xn′ be another random walk, independent of Sn, such that its characteristic function f′(t)=E[exp⁡itXi′] also satisfies the expansion (6). Then using [13, Lemma 3.1], one can adapt the proof of [13, Theorem 2.1] to show that Ln′(α)=Ln(α)+o(Ln(α)).


## 2.1. General Bounds

We first develop a technique to treat random walks with independent but not necessarily identically distributed increments.Proposition 4 (general upper bound).
Assume thatX1,X2,… are independent Zd-valued random variables and let Su,v≔Xu+⋯+Xu+v. Suppose further that for all n∈N and integers a,u,b,v≥0, with a+u≤b and any x∈Zd, one hasAPSa,u±Sb,v=x≤ϕu+v,BPSa,u=0-PSa,u+Sb,v=0≤ψu,v,where ϕ(u) is nonincreasing and ψ(u,v) is nonincreasing in u and is nondecreasing and subadditive in v in the sense that ψ(u,v+w)≤Aψ[ψ(u,v)+ψ(u,w)], for some constant Aψ independent of u, v, and w. Then, for some constant K=cAψ(1+Aψ)α-2 depending only on α(9)var⁡Lnα≤Kn∑i=0n-1ϕi2α-4∑i,j,k=0n-1ϕj∨iϕk∨i+ϕjψi+k,j.Proof of Proposition4.
We first write out the variance as a sum(10)varLnα=α!2∑k1≤⋯≤kα∑l1≤⋯≤lαPSk1=⋯=Skα,Sl1=⋯=Slα-PSk1=⋯=SkαPSl1=⋯=Slα.An important role is played by the manner in which the two sequences are interlaced, since, for example, if kα≤l1 or lα≤k1, the term vanishes by the Markov property.
We will treat the sum over indices withk1≤l1. The sum over the remaining index set with k1>l1 can be treated in a similar fashion and will contribute a constant factor. Therefore, we assume that k1≤l1 and we arrange the two sequences in an ordered sequence of combined length 2α which we denote as (p1,…,p2α); we also define (ϵ1,…,ϵ2α) where ϵi=0 if pi came from k≔{k1,…,kα} and ϵi=1 if pi came from l≔{l1,…,lα}. Finally we define two new sequences m0,m1,…,m2α-1, and δ1,…,δ2α-1, where m0≔p1,mi=pi+1-pi, and δi=ϵi+1-ϵi, for i=1,…,2α-1. Notice that since we assume that k1≤l1, we have p1=k1 and ϵ1=0. Let v(δ)≔∑i=12α-1|δi| denote theinterlacement index. The terms with v=1 vanish, while the terms with v=2 will be considered separately.
Terms with v ≥ 3. We first consider the sum In over the terms with v≥3 for which we drop the negative part and obtain the bound(11)In≔∑k1≤⋯≤kαl1≤⋯≤lαk1≤l1,vδ≥3PSk1=⋯=Skα,Sl1=⋯=Slα=∑x,y∈Zd∑p1≤⋯≤p2α≤n∑ϵ:vδ≥3PSp1=x,Sp2=x+ϵ2y,…,Sp2α=x+ϵ2αy≤∑x,y∈Zd∑m0,…,m2α-1≤n∑δ:vδ≥3PSm0=xPSm0,m1=δ1y⋯PSm2α-2,m2α-1=δ2α-1y=∑y∈Zd∑m0,…,m2α-1≤n∑δ:vδ≥3PSm0,m1=δ1y⋯PSm2α-2,m2α-1=δ2α-1y.Summing over the free index m0, it is clear that (12)In≤n+1∑m1,…,m2α-1∑y∈Zd∑δ:vδ≥3∏t=12α-1supw⁡PSw,mt=δty.For any δ=(δ1,…,δ2α-1) with v(δ)=v, exactly u≔2α-1-v elements are equal to 0, and therefore by Assumption A with x=0 we have (13)In≤Cn+1∑v=3α∑i=0nϕi2α-1-v∑j1,…,jv=0n∑y∈Zd∑δ′∈-1,+1v∏t=1vsupwt⁡PSwt,jt=δty.Letting (S~n)n∈N0 denote an independent copy of the random walk (Sn)n∈N0 and assuming without loss of generality that j1≤⋯≤jv, we have that for any δ∈{-1,+1}v(14)∑y∈Zd∏t=1vsupwt⁡PSwt,jt=δty≤∏t=2v-1supysupwt⁡PSwt,jt=ysupw1,wv⁡PSw1,j1-δvS~wv,jv=0≤∏t=2v-1ϕjtϕj1+jv≤∏t=2vϕjt∨j1. Let Gn≔∑i=0n‍ϕ(i). Since ϕ is nonincreasing we have that (15)Δn,v≔∑0≤j1≤⋯≤jv≤n∏t=2vϕjt∨j1≤∑jv=0nϕjv∑0≤j1≤⋯≤jv-1≤n∏t=2v-1ϕjt∨j1=GnΔn,v-1,and iterating this procedure, for v≥3, we have that Δn,v≤Δn,3Gnv-3. Combining the two bounds and summing over v=3,…,2α-1, we have that (16)In≤∑v=32α-1cαnGn2α-1-vΔn,v≤cαnGn2α-1-v+v-3Δn,3=cαnGn2α-4Δn,3,where c(α) is a constant depending only on α.
Terms with v = 2. Next we consider the sum Jn over the terms with v=2, which occurs when, for some j, the indices l1,…,lα all lie in [kj,kj+1]. Then it is easy to see that this sum Jn is bounded above by (17)Jn≤Cnsupw0,…,w2α-1∑mα+1,…,m2α-2=0n∏r=α+12α-2PSwr,mr=0∑m0,…,mα=0n∏t=1α-1PSwt,mt=0PSw0,m0+Swα,mα=0-PSw0,m0+⋯+Swα,mα=0≤CnGnα-2supw0,…,wα⁡∑m0,…,mα=0n∏t=1α-1PSwt,mt=0PSw0,m0+Swα,mα=0-PSw0,m0+⋯+Swα,mα=0≤CnGnα-2∑m0,…,mα=0n∏t=1α-1ϕmtψm0+mα,m1+⋯+mα-1≤CαnGnα-2Aψ1+Aψα-2∑m2,…,mα-1∏t=2α-1ϕmt∑m0,m1,mαϕm1ψm0+mα,m1≤CαAψ1+Aψα-2nGn2α-4∑i,j,k=0nϕjψi+k,j.The following corollary provides explicit bounds in the cases that are usually considered in the literature.Corollary 5.
Assume that the conditions of Proposition4 are satisfied with ϕ(m)=Tm-r and ψ(m,k)=Tm-r-1(k∧m). Then,(18)
v
a
r
L
n
α
≤
c
α
T
2
α
-
2
n
2
log
⁡
n
2
α
-
4
,
i
f
r
=
1
,
n
4
-
2
r
,
i
f
1
<
r
<
3
2
,
n
log
⁡
n
,
i
f
r
=
3
2
,
n
,
i
f
r
>
3
2
.It is straightforward to see that Corollary5 includes random walks with mean zero and finite second moment; for example, d=2 corresponds to r=1 and d=3 to r=3/2. Therefore several relevant results in [3, 7–13] are obtained as a special case of Corollary 5 and extended to the case of independent but not necessarily identically distributed variables, for example, by applying the local limit theorem, as conducted in [8].Also when the random walk incrementX is in the domain of attraction of the one-dimensional symmetric Cauchy law [13, 14] or in the case of planar random walk with second moments [3, 7–9, 11], it is well known that the conditions of Proposition 4 are satisfied with ϕ(m)=T/m and ψ(m,k)=Tm-2(k∧m).However, we can do better for symmetric variables and show that conditionA implies B, which together with the comparison technique motivates the following results. For a real number x, we write [x] for the integer part of x.Proposition 6 (bounds via comparison with characteristic function of symmetric random variables).
LetX1,X2,…, be independent Zd-valued random variables and let fit≔Eexp⁡(itXi). Assume that there exist a measurable function f:Γ→[0,1] and a positive nonincreasing sequence (ϕ(m))m∈N0, such that(19)1-fit≤Tft,fi±t≤ft,∫Γftmdt≤ϕm,for all integers i,m≥0, all t∈Γ, and some positive constant T. Then there exists another positive constant K=c(α,d,T) such that (20)varLnα≤Kn∑i=0n-1ϕi22α-4∑j=0njϕj2∑k=j2nϕk2.Proof of Proposition6.
Using the notation of Proposition4, for positive integers a,u,b,andv, with a+u≤b, ϵj=±1, and any x∈Zd(21)PSa,u+ϵ·Sb,v=x≤12πd∫Γ∏j∈a,a+u∪b,b+vfjϵjtdt≤12πd∫Γftu+vdt≤12πdϕu+v.
To findψ(u,v), notice that since f(t)≥0, (22)ϕm≥∫Γftm1-ftmdt=∑j=0m-1∫Γftm+j1-ftdt≥m∫Γft2m1-ftdt≕mQ2mwhence Q(m)≤2ϕ([m/2])/m. Therefore, (23)PSa,u=0-PSa,u+Sb,1=0=12πd∫Γ∏j=aa+ufjt1-fb+1tdt≤CT∫Γftu1-ftdt≤CTϕu/2u. A telescoping argument implies that (24)PSa,u=0-PSa,u+Sb,v=0≤CTϕu2vu. On the other hand for u≤v we can obtain a tighter bound through(25)PSa,u=0-PSa,u+Sb,v=0≤PSa,u=0≤ϕu. Combining the two bounds above it follows that B is satisfied with ψ(u,v)≔ϕu/2min⁡(u,v)/u. Thus all conditions of Proposition 4 are satisfied and the result follows.The following corollary allows for the case whereϕ(m) is regularly varying.Corollary 7.
Assume that the conditions of Proposition6 are satisfied with ϕ(m)=h(m)m-r, r≥1, where h(·) is slowly varying at ∞. Then,(26)
v
a
r
L
n
α
≤
K
Δ
n
α
,
ϕ
≤
c
α
T
2
α
-
2
n
2
∑
k
=
1
n
h
k
k
2
α
-
4
,
f
o
r
r
=
1
,
n
4
-
2
r
h
2
n
,
f
o
r
1
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r
<
3
2
,
n
∑
k
=
1
n
h
k
2
k
,
f
o
r
r
=
3
2
,
n
,
f
o
r
r
>
3
2
.Several results in [3, 7–13] 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. In addition, following [2], we can construct a one-dimensional symmetric random walk with characteristic function f(t)=1-c|t|1/r+o(|t|1/r), where r=2/d for d=2,3 and r=1/2 for d≥4, whose asymptotic behaviour is similar to that of genuinely d-dimensional random walk.The following example of genuinely2-dimensional recurrent walk with infinite variance was motivated by Spitzer [1, pp. 87].Example 8.
LetX1,X2,… be independent, identically distributed, Z2-valued random variables, such that P(|X1|=k)=c/(k3log⁡(k)g), for k≥4 and g∈[0,1). Let (Sn)n∈N0 be the corresponding random walk in Z2. Then we have(27)var⁡Lnα≤cn2max⁡log⁡ng,log⁡log⁡n2α-4log⁡n-21-g,for n≥10. Under these assumptions we have that PSn=0≤c/nlog⁡(n)1-g, which is in thecritical range, where the random walk is recurrent, without second moment. To see why, we note that by a lengthy but straightforward calculation it can be shown that the characteristic function of X satisfies (19) with (28)ϕn=cnlog⁡e∨n1-g,ft=exp⁡-At2ht2,wherehr≔1+log⁡1r+1-g.The sequence ϕ(m) is identified via Fourier inversion, polar coordinates, and a Laplace argument, (29)∫Γftndt≤c∫01exp⁡-nr1+log⁡1r1-g+Oe-n≤cnlog⁡e∨n1-g≕ϕn.


## 2.2. Bounds for Identically Distributed Variables

Proposition 9 (general upper bound for i.i.d.).
LetX,X1,X2,…, be independent, identically distributed, Zd-valued random variables. Suppose that for any x∈Zd and all positive integers a, u, b, and v, with a+u≤b, it holds that(30)PSa,u±Sb,v=x≤ϕu+v,where {ϕ(m)}m∈N0 is a nonincreasing sequence. Then for some constant K=c(α) we have that (31)var⁡Lnα≤Kn∑i=0n-1ϕi2α-4∑j=0njϕj∑k=jαn+1ϕkα.Proof of Proposition9.
By inspecting the proof of Proposition6, we notice that we only need to bound the term Jn. Consider typical ordering (32)0≤i1≤⋯≤ik≤j1≤⋯≤jα≤ik+1≤⋯≤iα≤n, and let us change variables to (m0,…,m2α) such that m0+⋯+m2α=n. Then the contribution to Jn is given by(33)∑m0,…,m2α∏j≠k,k+α1≤j≤2α-1PSmj=0PSmk+mk+α=0-PSmk+⋯+mk+α=0.We keep mj fixed for j≠α,k+α and we sum over m=mk+mk+α from 0 to some M=M(n,{mj}j≠k,k+α). Then for given mk+1,…,mk+α-1, the term in the sum is (34)∑m=0Mm+1PSm=0-PSm+q=0,where q≔mk+1+⋯+mk+α-1. Then since M≤n-q, it is an easy exercise to show that this sum is bounded above by (35)∑m=0Mm+1PSm=0-PSm+q=0≤∑m=0q-1m+1PSm=0+q1n-q≥q∑m=qn-qPSm=0≤∑m=0αm∗∧nm+1PSm=0+αm∗∑m=m∗nPSm=0,where m∗=max⁡mk+1,…,mk+α-1. The result follows by summing over all indices apart from m∗ and changing the order of summation.


## 2.3. Proofs of Main Results

Proof of Theorem1.
We apply a comparison argument found to be useful in many areas (e.g., Montgomery-Smith and Pruss [15], and Lefèvre and Utev [16]). More specifically we bound the quantity var⁡(Ln) by the corresponding quantity for the symmetrised random walk.
Following Spitzer’s argument we notice that withf(t)=E[exp⁡(it·X1)](36)PSa,u+ϵSb,v=x≤c∫Γftuf-tvdt=c∫Γft2u/2f-t2v/2dt.Since |f(t)|2 is the characteristic function of a symmetric random variable in Zd, for some positive λ, we have 1-|f(t)|2≥λ|t|2, and, hence, (37)PSa,u+ϵSb,v=x≤c∫Γexp⁡-λu+v2t2dt≤cu+v-d/2. The result follows from Proposition 9 applied with ϕ(m)=m-d/2.The proof of Theorem2 will be based on the following lemma.Lemma 10.
AssumeX,X1,X2,… are independent, identically distributed, genuinely d-dimensional random variables such that E|X|2=∞. Then there exists a monotone, slowly varying sequence (hn)n∈N0, such that hn→0 as n→∞ and(38)supx∈Zd⁡PSn=x≤cd∫ΓEeit·Xndt≤hnn-d/2.Proof of Lemma10.
Without loss of generality we assume thatX is symmetric. Let σe,L≔E[(e·X)21(|X|≤L)]. Following Spitzer, since X is genuinely d-dimensional, we may assume that there exist positive constants c,W, such that for any unit vector |e|=1 we have that σe,W≥c and 1-f(t)≥c|t|2 for all t∈Γ. Let λd be the d-dimensional Lebesgue measure on Rd and μd the Lebesgue-Haar measure on Sd-1≔{e∈Γ:|e|=1}. Notice that since E|X|2=∞, for any K, we have μd{e:σe,∞<K}=0.
Fix a small positivex such that c/x≥2W, and for any ϵ>0 let K=K(ϵ)=ϵ-d/2. Then there exists L=L(ϵ)>0 small enough so that μd{e:σe,L<K}≤ϵd/2. We partition Sd-1 in two sets (39)AL,K=e∈Sd:σe,L≥K,A¯L,K=e∈Sd:σe,L<K, so that, for any direction e∈A¯L,K, (40)z∈R:1-fze≤x⊆z:cz2≤x⊆z:z≤xc.Hence, using d-dimensional spherical coordinates, (41)λdz,e∈R×A¯L,K:1-fez≤x≤μdA¯L,Kxcd/21d≤ϵd/2xcd/21d. On the other hand, for any t, (42)1-ft=2∑k∈Zdsin⁡t·k22PX=k≥14Et·X2It·X≤12=t24σt/t,1/2t. Now, assume that c/x≥2L. Then for any direction e∈AL,K, by choice of x and since σe,L is increasing in L, for cz2≤1-f(ez)≤x or z≤x/c, it must be the case that (43)x≥1-fez≥z24σe,1/2z≥z24σe,L≥z24K,implying that, on the set AL,K, it must be that |z|≤2x/K. Changing to d-dimensional polar coordinates, we find that (44)λdz,e∈R×AL,K:1-fez≤x≤∫AL,K∫04x/Krd-1drde≤Cdϵd/2xd/2. Overall, for x≤c/4L2, λd{t:1-f(t)≤x}≤cd(xϵ)d/2, and hence {t∈Γ:1-f(t)≤x} has Lebesgue measure o(xd/2).
LetF(x) be the cumulative distribution function of the random variable log⁡(1/f(·)) defined on the probability space Γ with normalised Lebesgue measure. Then F is continuous at x=0 and supported on R+. Moreover, we have that F(x)=o(xd/2) as x↓0. Therefore, for some positive sequence (ϵn)n∈N0 with ϵn→0, we have that(45)12π2∫Γftndt=∫0∞e-nxdFx=n∫0∞e-nxFxdx≤n-d/2ϵn.
It remains to show that there exists a positive, monotone, slowly varying sequence(hn)n∈N0, such that ϵn≤h(n)→0 as n→∞. Let δn=supj≥nϵj and a0≔0 and for n≥1 define an recursively by an=min⁡(2a2r-1,1/δn), for 2r-1<n≤2r, so that an→∞ is monotone, a2r≤2a2r-1 implying that a2n≤4an, and 1/an≥δn≥ϵn. Finally, take hn≔1/max⁡(a0,log⁡an).Proof of Theorem2.
Assume thatE|X|2=∞ and d=2 or d=3. Then, by Lemma 10 there exists a slowly varying sequence hn→0 as n→∞ such that ∫Γ‍|Eexp⁡(it·X)|ndt≤hnn-d/2. Applying Corollary 7 with r=1 and r=3/2 we, respectively, find that (46)
var
⁡
L
n
α
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1
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2
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⁡
n
2
α
-
4
,
for
d
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K
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∑
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n
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ln
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for
d
=
3
.
Finally assume thatE|X|2<∞ and E[X]=μ≠0. Then P(Sn=0)=P(Sn′=-nμ) whence it follows that P(Sn=0)=o(n-d/2) (see, e.g., [17, Theorem 2.3.10]). Then inspecting the proof of Proposition 4, one can readily obtain the desired bound for the Jn term, while with slight modification the bound for the In term also follows.
Note that ford=1 the situation is much simpler since then var⁡(LnSRW(α))~C[ELnSRW(α,d)]2 and if E|X|2=∞ or E[X]≠0, ELnSRW(α,d)=o(n(1+α)/2).Proof of Theorem3.
We first give the proof for the cased=1. As in the proof of Proposition 4 we begin from expression (10) and define the sequences pi and δi for i=1,…,2α-1, and the quantity v(δ)=∑i=12α-1‍|δi|. Recall that v(δ) measures the interlacement of the two sequences k1,…,kα and l1,…,lα. For example, v(δ)=1 occurs when either kα≤l1 or lα≤k1, in which case the contribution vanishes by the Markov property. On the other hand v(δ)=2 when, for example, l1,…,lα∈[ki,ki+1] for some i. Finally v(δ)=3 occurs when, for example, (47)k1≤⋯≤kr≤l1≤⋯≤ls≤kr+1≤⋯≤kα≤ls+1≤⋯≤lα≤n.From the proof of Proposition 4, and using the bound P(Sn=0)≤c/n, the terms of the sum are bounded above by n2log(n)2α-1-v(δ), and thus the leading term appears when either v(δ)=2,3, with other terms giving strictly lower order. We will therefore analyze these two situations in detail in order to derive the exact asymptotic constants. When v=3, the two terms in the difference individually give the correct order and will be treated by the classical Tauberian theory. However for v=2, the two terms only give the correct order when considered together. This however forbids the use of Karamata’s Tauberian theorem since the monotonicity restriction would require roughly that Xi is symmetric. Thus the complex Tauberian approach, as developed in [13], is required to justify the answer.
Case  1 ( v ( δ ) = 3 ). Assume that part of the sequence l={l1,…,lα} lies between kr and kr+1 and the rest between ks and ks+1. Then using the change of variables (48)i1=m0,i2=m0+m1,⋮ir=m0+⋯+mr-1j1=m0+⋯+mr,j2=m0+⋯+mr+1,⋮js=m0+⋯+mr+s-1,ir+1=m0+⋯+mr+s,ir+2=m0+⋯+mr+s+1,⋮iα=m0+⋯+mα+s-1,js+1=m0+⋯+mα+s,js+2=m0+⋯+mα+s+1,⋮jα=m2α-1,n=m0+⋯+m2α,we rewrite the positive term in (10) as (49)an=∑PSi1=⋯=Siα;Sj1=⋯=Sjα=∑m0,…,m2α-1∏j=1j≠r,r+s,α+s2α-1PSmj=0PSmr+Smr+s′=Smr+s′+Smα+s′′=0. Notice that from [13] we have that ∑n≥0‍λnP(Sn=0)~log⁡(1/(1-λ))/πγ. Let (50)aλ=1-λ-3-log⁡1-λ2α-4,cγ=πγ-2α+4.Then, by direct calculations and Fourier inversion formula (51)∑n≥0λnan=cγ1-λaλ∑x∈Z∑k1,k2,k3≥0λk1+k2+k3PSk1=xPSk2=-xPSk3=x=cγ1-λaλ12π2∬Γdtds1-λft1-λfs1-λft+s~cγ1-λaλ12π2γ211-λ∬R2dxdy1+x1+y1+x+y~14γ2cγaλ. Next we consider the negative term in (10) (52)bn≔∑m0,…,m2α-1PSm1=⋯=Smr-1=Smr+⋯+Smr+s=Smr+s+1=⋯=Smα+s-1=0PSmr+1=⋯=Smr+s+⋯+Smα+s=Smα+s+1=⋯=Sm2α-1=0. By direct calculations and (6),(53)∑nλnbn=1πγlog⁡11-λα-s+r-21-λ-2∑mr,…,mα+s=0∞λmr+⋯+mα+s∏t=r+1,…,α+s-1t≠r+sPSmt=0PSmr+⋯+Smr+s=0PSmr+s+⋯+Smα+s=0, and using Fourier inversion and (6) the internal sum behaves as (54)2π-α-s+r∫-ππ⋯∫-ππ1-λϕx-11-λϕxϕy-11-λϕy-1∏j=r+1r+s-1∏k=r+s+1α+s-11-λϕxϕtj-11-λϕyϕtk-1dtjdtkdxdy~πγ-α-s+r1-λ-1log⁡11-λα-r+s-2π26.Then, we have ∑n‍λnb(n)~(π2/6(πγ)2α-2)a(λ), whence the Tauberian theorem implies that a(n)-b(n)~n2log(n)2α-4/24π2α-4γ2α-2. Most importantly we see that the lengths and locations of the chains, r and s, do not affect the asymptotic behaviour. Noting that if 1≤r, s≤α-1, we can partition 2α=r+s+(α-r)+(α-s) in (α-1)2 ways, and thus overall the total contribution from terms with v=3 is (55)α!α-1212π2α-4γ2α-2n2logn2α-4.
Case  2 ( v ( δ ) = 2 ). The typical term c(n) was introduced in (33) in the proof of Proposition 9. Now we let λ∈C, with |λ|<1. By lengthy but direct calculations we can derive an expression of the form (56)∑nλncn=α-1γπ2α-2aλ+oaλ,λ⟶1.The approach developed in [13] can then be used to bound the error terms and show that c(n)~[(α-1)/2(γπ)2α-2]n2log(n)2α-4.
Finally taking into account the fact thatl1,…,lα can be in any of the α-1 intervals [ki,ki+1], for i=1,…,α-1, the result follows the overall contribution of terms with v(δ)=2(57)α-122γπ2α-2n2log⁡n2α-4.
The case ford=2 is very similar, so we move on to the case d=3.
Case  3 ( d = 3 and α = 2 ). Using the same notation as before, we have three terms to consider a(n), b(n), and c(n). We first consider c(n). Letting K≔ϵ/1-λ and using the usual power series construction and spherical coordinates(58)∑nλncn=1-λ-22π-6∬J3×J3λfy1-fxdxdy1-λfx21-λfy1-λfxfy~22π-4Σ-11-λ-2∬0Kr4s2drds1+r221+s21+r2+s2~22π-4Σ-1π21-λ-2log⁡11-λ≕κ11-λ-2log11-λ,and thus cn~κ1nlog⁡n, where κ1>0, where the answer can be justified following [13].
The terma(n)-b(n) is trickier to compute. As usual we consider the power series (59)∑n≥0λnan-bn=1-λ-22π-6∬Bϵdxdy1-λfx1-λfy1-λfx+y-1-λ-22π-6∬Bϵdxdy1-λfx1-λfy1-λfxfy=1-λ-22π-6I1λ-I2λ. Let A∈[-1,1] be the cosine of the angle between x and y, which in spherical coordinates is (60)A=Aθ1,θ2,ϕ1,ϕ2=cos⁡ϕ1-ϕ2sin⁡θ1sin⁡θ2+cos⁡θ1cos⁡θ2.Then as 0<λ↑1, using the expansion (6) (61)I1λ~Σ-1∫r,s=0ϵ∫ϕ1,2=02π∫θ1,θ2=0πr2s2sin⁡θ1sin⁡θ2dθ1dθ2dϕ1dϕ2drds1-λ+λr21-λ+λs21-λ+λr2+s2+2Ars=Σ-1∫θ1,θ2=0π∫ϕ1,ϕ2=02π∫r,s=0Ksin⁡θ1sin⁡θ2r2s2dsdrdϕ1dϕ2dθ1dθ21+r21+s21+r2+s2+2Ars~Σ-1log⁡K∫θ1,θ2=0π∫ϕ1,ϕ2=02πsinθ1sinθ2arccos⁡Aθ1,θ2,ϕ1,ϕ21-Aθ1,θ2,ϕ1,ϕ22dϕ1dϕ2dθ1dθ2.The other integral is slightly easier (62)I2λ~Σ-1π2log⁡K∫θ1,θ2=0π∫ϕ1,ϕ2=02πsin⁡θ1sin⁡θ2dϕ1dϕ2dθ1dθ2,and thus overall we must have that(63)I1-I2λ~122π-6Σ-11-λ-2log⁡11-λ∫θ1,θ2=0π∫ϕ1,ϕ2=02πarccos⁡A1-A2-π2sin⁡θ1sin⁡θ2dϕ1dϕ2dθ1dθ2≕κ21-λ-2log⁡11-λ,whence it follows that var⁡(Ln(2))~κ1+κ2nlog⁡n.
To prove the last claim letSn′=X1′+⋯+Xn′ be another random walk, independent of Sn, such that its characteristic function f′(t)=E[exp⁡itXi′] also satisfies the expansion (6). Then using [13, Lemma 3.1], one can adapt the proof of [13, Theorem 2.1] to show that Ln′(α)=Ln(α)+o(Ln(α)).



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*Source: 5370627-2016-07-20.xml*