# Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

Authors: Wensheng Wang; Anwei Zhu
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

LetX={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi(k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)(n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.


---

## Body

## 1. Introduction

LetX={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi(k≥1). Let σ={σ(x),x∈Z} be a sequence of R-valued random variables which are independent of X’s. We refer to S={Sk,k≥0} as therandom walk and σ as therandom scenery. Then the process K={Kn,n∈N} is defined by (1)Kn=∑k=0nσSk,n∈N,where N={0,1,2,…} and ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1, called arandom walk in random scenery (RWRS, in short), sometimes also referred to as theKesten-Spitzer random walk in random scenery; see Kesten and Spitzer [1]. An interpretation is as follows. If a random walker has to pay σ(x) units at any time he/she visits the site x, then Kn is the total amount he/she pays by time n.RWRS was first introduced by Kesten and Spitzer [1] and Borodin [2, 3] in order to construct new self-similar stochastic processes. Kesten and Spitzer [1] proved that when the random walk and the random scenery belong to the domains of attraction of different stable laws of indices 1<α≤2 and 0<β≤2, respectively, then there exists δ>1/2 such that {n-δK⌊nt⌋,t≥0} converges weakly as n→∞ to a continuous δ-self-similar process with stationary increments, δ being related to α and β by δ=1-α-1+(αβ)-1. The limiting process can be seen as a mixture of β-stable processes, but it is not a stable process. When 0<α<1 and for arbitrary β, the sequence {n-1/βK⌊nt⌋,t≥0} converges weakly, as n→∞, to a stable process with index β (see Castell et al. [4]). Bolthausen [5] (see also Deligiannidis and Utev [6]) gave a method to solve the case α=1 and β=2 and, especially, he proved that when S is a recurrent Z2-random walk, the sequence {(nlog⁡n)-1/2K⌊nt⌋,t≥0} satisfies a functional central limit theorem. More recently, the case S one- or two-dimensional random walks and β∈(0,2) was solved in Castell et al. [4]; the authors prove that the sequence {n-1/β(log⁡n)1/β-1K⌊nt⌋,t≥0} converges weakly to a stable process with index β. Finally for any arbitrary transient random walk, it can be shown that the sequence {n-1/2Kn,n∈N} is asymptotically normal (see for instance Spitzer [7] page 53).Among others, we can cite strong approximation results [8–10], laws of the iterated logarithm [11–13], limit theorems for correlated sceneries or walks [14–17], large and moderate deviations results [18–22], and ergodic and mixing properties (see the survey [23]).The problem we investigate in the present paper has already been studied in Lewis [24] in the case that random sceneries σ’s satisfy E[σ(0)]=0 and E[σ2(0)]=1, and the random walk S (which can be Zd-valued) satisfies some mild conditions. Lewis [24] established the following LIL:(2)limsupn→∞⁡V2,n-1Kn2log⁡log⁡n=1a.s.,where ξn(x) is the number of visits of the random walk to the point x∈Z in the time interval [0,n], i.e.,(3)ξnx≔#0≤k≤n:Sk=x=∑k=0nISk=x,n≥0,x∈Z.Here and in the sequel, the following notation is used: for a>0 and n≥0,(4)Va,n=∑x∈ZEξnax1/a.It is therefore natural to investigate limit behavior of RWRSKn when the sceneries σ’s do not have finite second moment. For the sake of convenience, we are summarizing here the main assumptions we are making on the sceneries σ’s. Assume that the sceneries σ’s belong to the domain of attraction of a stable law Gβ(0<β<2); that is, σ’s satisfy that(5)limn→∞⁡Pn-1/β∑k=1nσk≤x=Gβx,where Gβ is a stable distribution of index 0<β<2, with characteristic function(6)exp⁡-θβA1+iA2sgn⁡θfor some 0<A1<∞, A1-1A2≤tan⁡(π/2)β. From the known characterization of the domain of attraction of a stable law Gβ (Feller [25], II, Chap. 17) it follows that, for 0<β<2, (5) and (6) are equivalent to(7)Pσ0≥x~c1,1xβ,Pσ0≤-x~c1,2xβas x→∞ for suitable constants c1,1 and c1,2. Note that (5) and (6) imply(8)Eσ0=0ifβ>1.For β=1 we impose an additional condition (stronger than (5) and (6)), namely, that for some positive constant c0,(9)Eσ0Iσ0≤ρ≤c0<∞∀ρ>0.It is well known that LILs for heavy tailed random variables are different from those for random variables attracted to the normal law. We have to use power norming and the resulting limit theorem is called Chover-type LIL (see Chover [26]). The main results of this paper read as follows.Theorem 1.
Letσ={σ(x),x∈Z} be a sequence of i.i.d. random variables satisfying (5) and (9), and X={Xi,i≥1} be a sequence of i.i.d. random variables with a common distribution F and independent of σ’s. Assume that F is supported on [0,∞), absolutely continuous, and 1-F(x)~c1,1x-α, 0<α<2. Then(10)limsupn→∞⁡Vβ,n-1Kn1/log⁡log⁡n=e1/βa.s.Theorem1 gives the following information about the maximal growth rate of RWRS Kn.Corollary 2.
We have for allε>0, with probability one,(11)Kn>Vβ,nlog⁡n1+ε/βfor  at  most  finitely  manynand(12)Kn>Vβ,nlog⁡n1-ε/βfor  infinitely  manyn.Remark 3.
It follows from Corollary2 that the maximal growth rate of Kn is of the order Vβ,n(log⁡n)1/β. Equation (10) is equivalent to (11) and (12). In fact, (11) implies that (13)log⁡Vβ,n-1Kn-1+εβlog⁡log⁡n≤0a.s.for all large n. Letting ε↓0, it yields that the limit superior on left-hand side of (10) is less than e1/β. Equation (12) implies that (14)log⁡Vβ,n-1Kn-1-εβlog⁡log⁡n≥0a.s.for infinitely many n. Letting ε↓0, it yields that the limit superior on left-hand side of (10) is greater than e1/β. Moreover, from the proof of Theorem 1 below, the upper bound of (10) does not need the assumptions that F is supported on [0,∞) and absolutely continuous.Complementary to Theorem1 we have the following clustering statement, which gives additional information about the path behavior of RWRS Kn.Theorem 4.
Under the assumptions of Theorem1, with probability one, every point in the interval (1,e1/β] is a cluster point of the sequence:(15)Vβ,n-1Kn1/log⁡log⁡n:n≥1.Throughout this paper, we use the notations:an=o(bn) if an/bn→0, an~bn if lim⁡an/bn=1 and an≍bn if c1,2≤liminf⁡an/bn≤limsup⁡an/bn≤c1,3. Let i.o. mean infinitely often, a.s. mean almost surely, E[·] mean expectation, and EF[·] mean conditional expectation given σ-field F. An unspecified positive and finite constant will be denoted by c, which may not be the same in each occurrence. More specific constants in Section i are numbered as ci,1,ci,2,…. The sign ⌊·⌋ sometimes denotes the integer part anf at other times denotes usual brackets; it will be clear from the context. Since we shall deal with index n which ultimately tends to infinity, our statements, sometimes without further mention, are valid only when n is sufficiently large.


## 2. Preliminaries

In this section we investigate some technical results necessary for our argumentation. We will first present a version of the Borel-Cantelli lemma to sums of conditional probabilities (see, e.g., Theorem 2.8.5 in Stout [27]).Lemma 5.
Let{En,n≥1} be a sequence of arbitrary events and {Gn,n≥1} be an increasing sequence of σ-fields such that En∈Gn for each n≥1. Then(16)Eni.o.=∑n=1∞PEn∣Gn-1=∞,that is, ∑n=1∞PEn∣Gn-1<∞ implies that En occur at most finitely often and ∑n=1∞PEn∣Gn-1=∞ implies that En occur infinitely often.We will need the following large deviation inequalities for RWRS, which may be of independent interest.Lemma 6.
Let{σ(x),x∈Z} be a sequence of i.i.d. random variables satisfying (5) and (9), and {Xi,i≥1} be a sequence of arbitrary random variables and independent of σ’s. Let {tn,n≥1} be a sequence of positive numbers such that tn→∞. Then (17)0<liminfn→∞⁡tnβPKn≥tnVβ,n≤limsupn→∞⁡tnβPKn≥tnVβ,n<∞.Proof.
We denote byF=σ(X1,X2,…) the σ-field generated by the random walk and (18)η~nx=ξnxσxIσx≤tnVβ,nξn-1x,n≥0,x∈Z.
By (7), for all u>0 and 0<β<2, (19)c2,1u-β≤Pσ0>u≤c2,2u-β.Thus, (20)∑x∈ZPσx≥tnVβ,nξn-1x=∑x∈ZEPσx≥tnVβ,nξn-1x∣F≤c2,3tn-β.By (19), for all x∈Z, (21)EFη~nx2≤EFξn2xσ2xIσx≤tnVβ,nξn-1x=ξn2x∫0tnVβ,nξn-1xuPσx≥udu≤c2,4ξn2x∫0tnVβ,nξn-1xu1-βdu≤c2,5tn2-βVβ,n2-βξnβx.It follows that (22)Eη~nx2≤c2,5tn2-βVβ,n2-βEξnβx.On the other hand, if β∈(0,1), (23)EFη~nx≤EFξnxσxIσx≤tnVβ,nξn-1x=ξnx∫0tnVβ,nξn-1xPσx>udu≤c2,6ξnx∫0tnVβ,nξn-1xu-βdu≤c2,7tn1-βVβ,n1-βξnβxfor all x∈Z; if β=1, by (9), (24)Eη~nx≤EEFη~nx≤c2,8Eξnxfor all x∈Z; and, if β∈(1,2), by (8) and (19), (25)EFη~nx≤EFξnxσxIσx≥tnVβ,nξn-1x=ξnx∫tnVβ,nξn-1x∞Pσx>udu≤c2,9ξnx∫tnVβ,nξn-1x∞u-βdu≤c2,10tn1-βVβ,n1-βξnβxfor all x∈Z. Hence, by (23)-(25) and making use of the fact that E[η~n(x)]=E[EF[η~n(x)]], we have (26)tnVβ,n-1∑x∈ZEη~nx→0asn→∞.Thus, by (22) and (26), (27)P∑x∈Zη~nx≥tnVβ,n≤P∑x∈Zη~nx-Eη~nx≥tnVβ,n2≤4tnVβ,n-2∑x∈ZEη~nx-Eη~nx2≤4tnVβ,n-2∑x∈ZEη~nx2≤c2,11tn-β.
Noting that we can rewriteKn as (28)Kn=∑x∈Zξnxσx,we have that (29)PKn≥tnVβ,n≤∑x∈ZPσx≥tnVβ,nξn-1x+P∑x∈Zη~nx≥tnVβ,n.It follows from (20), (27), and (29) that (30)PKn≥tnVβ,n≤c2,12tn-β.By replacing σ(x) with -σ(x), we have (31)P-Kn≥tnVβ,n≤c2,13tn-β.This, together with (30), yields (32)PKn≥tnVβ,n≤c2,14tn-β.It yields the right-hand side of (17).
To verify the left-hand side of (17), we denote by Ax and Bx the events {ξnxσx≥(1+ε)tnVβ,n} and {∑y∈Z,y≠xξnyσy<εtnVβ,n}, ε>0,x∈Z, respectively. By (19) and some conditional argument, we have (33)∑x∈ZPAx≍tn-β→0.On the other hand, by (28), (34)PBx≥PKn≤ε2tnVβ,n-Pξnxσx≥ε2tnVβ,n.This, together with (20) and (32), yields that (35)PBx→1∀x∈Z.Note that (36)PKn>tnVβ,n≥EP⋃x∈ZAx∩Bx∣F=ElimM→∞⁡P⋃x=-MMAx∩Bx∣F≥ElimM→∞⁡∑x=-MMPAx∩Bx∣F-∑-M≤x<y≤MPAx∩Bx∩Ay∩By∣F≥∑x∈ZPAxBx-∑-∞≤x<y≤∞PAxAy=∑x∈ZPAxPBx-∑x∈ZPAx2.Thus, by (33)-(36), (37)PKn>tnVβ,n≥c2,15∑x∈ZPAx≥c2,16tn-β.It yields the left-hand side of (17). The proof of Lemma 6 is completed.We will also need the following two technical results.Lemma 7.
Let{Xi,i≥1} be a sequence of i.i.d. nonnegative random variables with a common distribution F. Assume that F is absolutely continuous and 1-F(x)~c2,17x-α, 0<α<2. Then, for all r>0, (38)Sn≥n1/αlog⁡n-ra.s.Proof.
LetMn=max⁡{X1,…,Xn}, F¯(x)=1-F(x)=P(X1>x), and F¯∗ be the inverse of F¯. Let U,U1,U2,…,Un be i.i.d. random variables with the distribution of U uniform over (0,1) and Mn∗=max⁡{U1,U2,…,Un}. Let θ>1 be a constant which will be chosen later on and nk=⌊θk⌋, k≥1. By making use of the fact that F(Xn) is a uniform (0,1) random variable, we have Mn∗=dF(Mn), n≥1. On the other hand, F¯∗(y)~c2,18y-1/α, 0<y≤1. Thus, since Xi’s are nonnegative, F¯ and F¯∗ are nonincreasing: (39)PSnk≤nk1/αlog⁡nk-r≤PMnk≤nk1/αlog⁡nk-r≤PF¯∗F¯Mnk≤F¯∗c2,19nk-1log⁡nkαr=PF¯Mnk≥c2,19nk-1log⁡nkαr=P1-Mnk∗≥c2,19nk-1log⁡nkαr=PMnk∗≤1-c2,19nk-1log⁡nkαr=PU≤1-c2,19nk-1log⁡nkαrnk≤exp⁡-c2,19log⁡nkαr.By making use of Borel-Cantelli lemma, (40)liminfk→∞⁡nk-1/αlog⁡nkrSnk≥1a.s.To each n, there exists an integer k such that nk≤n≤nk+1. Thus, by (40), (41)liminfn→∞⁡n-1/αlog⁡nrSn≥liminfk→∞⁡minnk≤n≤nk+1⁡n-1/αlog⁡nrSn≥liminfk→∞⁡nk1/αnk+1-1/αnk-1/αlog⁡nkrSnk≥θ-1/αa.s.Letting θ↓1, (38) is proved. The proof of Lemma 7 is completed.Lemma 8.
Let{σ(x),x∈Z} be a sequence of i.i.d. random variables satisfying (5) and (9), and {Xi,i≥1} be a sequence of arbitrary random variables and independent of σ’s. Let {an,n≥1} be a nondecreasing sequences of positive integers such that an→∞, Wn=∑x=-ananξn(x)σ(x) and γn=(∑x=-ananE[ξnβ(x)])1/β. Then (42)limsupn→∞⁡γn-1Wn1/log⁡log⁡n≤e1/βa.s.Proof.
Letn0=0 and nk=inf⁡{n:γn≥2k}(k≥1). Since γn is increasing and γn→∞, we have that nk→∞ and 2k≤γnk<2k+1. Noting (43)γn≤∑x∈Zξn2x1/β≤∑x∈Zξnx2/β=n2/β,we have (44)nk≥γnkβ/2≥2βk/2.
For the sake of convenience, we denoteuk=γnk(log⁡nk-1)(1+ε)/β, Sn={x∈Z:x∈[-an,an]}, (45)ηnx=ηnk,x=ξnxσxIσx≤ukξnk-1xif0<β≤1,ξnxmin⁡σx,ukξnk-1xif1<β<2,and W~n=∑x∈Snηn(x) for ε>0, x∈Z and nk-1<n≤nk.
By (19) and (44), (46)Pσx>ukξnk-1x∣F≤c2,20k-1+εγnk-βξnkβx.It follows that (47)∑x=-ankankPσx>ukξnk-1x≤c2,21k-1+ε.Since(48)min⁡σx,ukξnk-1x=σxIσx≤ukξnk-1x+ukξnk-1xIσx>ukξnk-1x,following the same argument as the proof of (29), we have (49)uk-1maxnk-1<n≤nk⁡EW~n∣F≤uk-1maxnk-1<n≤nk⁡∑x=-ananEηnx∣F→0as k→∞ for 0<β<2. On the other hand, by (22) and noting (50)Eηn2x∣F≤2Eξnk2xσ2xIσx≤ukξnk-1x∣F+2uk2Pσx≥ukξnk-1x∣F,we have for nk-1<n≤nk and 0<β<2, (51)Eηnx2∣F≤c2,22uk2-βξnkβx.It follows that (52)∑x∈SnkEηnkx2∣F≤c2,23uk2-βγnkβ.
From Newman and Wright [28], we call a finite collection of random variables Zi, 1≤i≤m, which is associated if any two coordinatewise nondecreasing functions f1,f2 on Rm such that Fi=fi(Z1,…,Zm) have finite variance for i=1,2, cov(F1,F2)≥0; an infinite collection is associated if every finite subcollection is associated. It is not difficult to demonstrate that independent variables are always associated. Moreover, given F, ηn(x)-E[ηn(x)∣F] are nonincreasing functions on σ and are also associated variables by Esary et al. [29]. Consequently, by Theorem 2 of Newman and Wright [28] and (52), (53)Emaxnk-1<n≤nk⁡W~n-EW~n∣F2∣F≤EW~nk-EW~nk∣F2∣F≤∑x∈SnkEηnkx2∣F≤c2,23uk2-βγnkβ.Hence (54)uk-2Emaxnk-1<n≤nk⁡W~n-EW~n∣F2∣F≤c2,24k-1+ε.Note that (55)Pmaxnk-1<n≤nk⁡Wn≥uk≤∑x=-ankankPσxξnkx>uk+Pmaxnk-1<n≤nk⁡W~n≥uk≤∑x=-ankankPσx>ukξnk-1x+EPmaxnk-1<n≤nk⁡W~n-EW~n∣F≥2-1uk∣F≤∑x=-ankankPσx>ukξnk-1x+Euk-2Emaxnk-1<n≤nk⁡W~n-EW~n∣F2∣F.Thus, by (47), (54), (55) and making use of Borel-Cantelli lemma, (56)limsupk→∞⁡maxnk-1<n≤nk⁡Wnγnk-1log⁡nk-1-1+ε/β≤c2,25a.s.By replacing σ(x) with -σ(x), following the same argument, we have that (56) also holds if Wn is replaced with -Wn. It yields (57)limsupk→∞⁡maxnk-1<n≤nk⁡Wnγnk-1log⁡nk-1-1+ε/β≤c2,26a.s.Therefore, by (57), (58)limsupn→∞⁡Wnγn-1log⁡n-1+ε/β≤limsupk→∞⁡maxnk-1<n≤nk⁡Wnγn-1log⁡n-1+ε/β≤c2,27limsupk→∞⁡maxnk-1<n≤nk⁡Wnγnk-1log⁡nk-1-1+ε/β≤c2,28a.s.It follows that (59)log⁡Wnγn-1-1+2εβlog⁡log⁡n≤0a.s.Letting ε↓0, we obtain (49). The proof of Lemma 8 is completed.


## 3. Proofs

Proof of Theorem1.
Letε∈(0,1) and τ∈(0,1/α) be two arbitrary constants. Let Sn, Wn, and γn be defined as in Lemma 8 with an=n1/α(log⁡n)1/α+τ. By Chover’s law of the iterated logarithm (see Chover [26] and Qi and Cheng [30]) we have Sn=o(an) a.s. Thus, for any sample point ω for which it holds, there exists N0=N0(ω) such that for all n≥N0 and x≥Card(Sn), where Card(Sn) is the number of integers belonging to Sn, ξn(x)(ω)=∑i=0nI[Si=x](ω)=0. It follows that, for all n≥N0,(60)∑x∉Snσxξnx=0a.s.,∑x∉Snξnβx=0a.s.By (28) and (60), we have that Kn=Wn a.s. and Vβ,n=γn a.s. Hence, to prove (10), by (28), it suffices to prove that, for all 0<ε<1,(61)limsupn→∞⁡Wnγn-1log⁡n-1+ε/β≤c3,1a.s.and(62)limsupn→∞⁡Wnγn-1log⁡n-1-ε/β≥c3,2a.s.By (42), (61) holds. Thus, it remains to prove (62). Let Tn={x∈Z:x∈[0,an]}. Let θ>1 be a constant. For k≥1, let mk=ekθ and nk=inf{n:n∈{m1,m2,…},γn≥ekθ}. Since γn is increasing and γn→∞, we have that nk’s are well-defined, nk→∞ and ekθ≤γnk<e(k+1)θ. Noting nβ~(∑x∈Tnξn(x))β≤γnβ if 0<β≤1 and n~∑x∈Tnξn(x)≤γnβ if 1<β<2, we have, for 0<β<2,(63)nk≤γnkβ∨1≤eβ∨1k+1θ.
Fork≥1, we denote λ1,k≔∑x∈Tn2k+1∖Tn2k-1E[ξn2k+1β(x)], λ2,k≔∑x∈Tn2k-1E[ξn2k+1β(x)], and vk≔λ1,k1/β(log⁡n2k+1)(1-ε)/β. Denote by Cx, Dx, and Ex the events Cx={σxξn2k+1x≥(1+ε)vk}, Dx={∑y∈Tn2k+1∖Tn2k-1,y≠xσyξn2k+1y<εvk}, and Ek=∑x∈Tn2k+1∖Tn2k-1σxξn2k+1x≥vk. Let Gk be the σ-field generated by {σ(x),x∈Tn2k+1}∪F. Then, Ek∈Gk. By (19) and (63),(64)∑x∈Tn2k+1∖Tn2k-1PCx∣Gk-1≥c3,3log⁡n2k+1-1+ε≥c3,4k-1-εθ.Similar to (36), we have(65)PEk∣Gk-1≥P⋃x∈Tn2k+1∖Tn2k-1Cx∩Dx∣Gk-1≥∑x∈Tn2k+1∖Tn2k-1PCx∣Gk-1PDx∣Gk-1-∑x∈Tn2k+1∖Tn2k-1PCx∣Gk-1.Similar to (33) and (35), we have ∑x∈Tn2k+1∖Tn2k-1P(Cx∣Gk-1)→0 and P(Dx∣Gk-1)→1 for all x∈Tn2k+1∖Tn2k-1, respectively. Thus, by (64) and (65), we have(66)PEk∣Gk-1≥c3,5∑x∈Tn2k+1∖Tn2k-1PCx∣Gk-1≥c3,6k-1-εθ.By choosing θ>1 small enough such that (1-ε)θ<1, and making use of Lemma 5,(67)PEki.o.=1.
By the definitions ofmk and nk, we have that(68)γn2kγn2k+1≤e2kθe2k+1θ≤e-2kθ-1→0,and that there exist integers j<l such that n2k-1=mj and n2k=ml. It follows that(69)n2k-1log⁡n2k-12ατn2k=j2ατθejθelθ→0.
By Lemma7, we have almost surely Sn>n1/α(log⁡n)-τ for all large n. This, together with (60), (68), and (69), yields(70)λ2,k=∑x∈Tn2k-1Eξn2k-1log⁡n2k-12ατβx≤γn2kβ=oγn2k+1β.Since γn2k+1β=λ1,k+λ2,k, by (70), we have γn2k+1β~λ1,k. Thus, (67) remains true when λ1,k is replaced with γn2k+1β. Hence, we have(71)P∑x∈Tn2k+1∖Tn2k-1σxξn2k+1x≥γn2k+1log⁡n2k+11-ε/βi.o.=1.
By (68), (70) and following the same argument as the proof of (32), (72)P∑x∈Tn2k-1σxξn2k+1x≥c3,7γn2k+1log⁡n2k+11-ε/β=P∑x∈Tn2k-1σxξn2k+1x≥c3,8λ2,kγn2k+1λ2,klog⁡n2k+11-ε/β≤P∑x∈Tn2k-1σxξn2k+1x≥c3,9e2kθ-1λ2,k≤c3,10e-β2kθ-1.Thus, by making use of Borel-Cantelli lemma,(73)P∑x∈Tn2k-1σxξn2k+1x≥c3,7γn2k+1log⁡n2k+11-ε/βi.o.=0.Noting (74)Wn2k+1=∑x∈Tn2k+1∖Tn2k-1σxξn2k+1x+∑x∈Tn2k-1σxξn2k+1x,by (71) and (73), (75)PWn2k+1≥c3,11γn2k+1log⁡n2k+11-ε/βi.o.=1.This yields (62). The proof of Theorem 1 is completed.Proof of Theorem4.
Fix0<λ≤1/β. Let ν=1/βλ and nk=ekν(k≥1). It is enough to prove that(76)limsupk→∞⁡Vnk-1Knk1/log⁡log⁡nk=eλa.s.
To prove (76), it suffices to prove that, for all ε>0, with probability one,(77)Vnk-1Knk>log⁡nk1+ελfor  at  most  finitely  manykand(78)Vnk-1Knk>log⁡nk1-ελfor  infinitely  manyk.
By Lemma6, (79)PVnk-1Knk>log⁡nk1+ελ≤c3,12log⁡nk-1+εβλ≤c3,13k-1+ε.By making use of Borel-Cantelli lemma, we obtain (77).
It remains to prove (78). For the case λ=1/β, following the same lines as the proof of (62), we have that there exists a subsequence of the subsequence {ek,k≥1} such that (78) holds. For the case 0<λ<1/β, we have ν>1. For k≥1, let mk=inf⁡{n:n∈{n1,n2,…},Vn≥ekν}. Following the same lines as the proof of (62), we have, with probability one, (80)Vmk-1Kmk>log⁡mk1-ελfor  infinitely  manyk.On the other hand, {mk} is a subsequence of the subsequence {nk}. Thus, we obtain (78) again. The proof of Theorem 4 is completed.



---

*Source: 8968947-2018-08-01.xml*