Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Abstract

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

1. Introduction

Let {𝑌_𝑖,𝐽_𝑖} be a sequence of independent and identically distributed random vectors, and write 𝑆(𝑛)=𝑌₁+𝑌₂+⋯+𝑌_𝑛 and 𝑇(𝑛)=𝐽₁+𝐽₂+⋯+𝐽_𝑛. Let 𝑁_𝑡=max{𝑛≥0∶𝑇(𝑛)≤𝑡} the renewal process of 𝐽_𝑖. A continuous time random walk (CTRW) is defined by 𝑋(𝑡)=𝑆𝑁_𝑡=^𝑁𝑡_𝑖=1𝑌_𝑖.(1.1) In this setting, 𝑌_𝑖 represents a particle jump, and 𝐽_𝑖>0 is the waiting time preceding that jump, so that 𝑆(𝑛) represents the particle location after 𝑛 jumps and 𝑇(𝑛) is the time of the 𝑛th jump. Then 𝑁_𝑡 is the number of jumps by time 𝑡>0, and the CTRW 𝑋(𝑡) represents the particle location at time 𝑡>0, which is a random walk subordinated to a renewal process.

It should be mentioned that the subordination scheme of CTRW processes is going back to Fogedby [1] and that it was expanded by Baule and Friedrich [2] and Magdziarz et al. [3]. It should also be mentioned that the theory of subordination holds for nonhomogeneous CTRW processes, that were introduced in the following works: Metzler et al. [4, 5] and Barkai et al. [6].

The CTRW is useful in physics for modeling anomalous diffusion. Heavy-tailed particle jumps lead to superdiffusion, where a cloud of particles spreads faster than the classical Brownian motion, and heavy-tailed waiting times lead to subdiffusion. CTRW models and the associated fractional diffusion equations are important in applications to physics, hydrology, and finance; see, for example, Berkowitz et al. [7], Metzler and Klafter [8], Scalas [9], and Meerchaert and Scalas [10] for more information. In applications to hydrology, the heavy tailed particle jumps capture the velocity irregularities caused by a heterogeneous porous media, and the waiting times model particle sticking or trapping. In applications to finance, the particle jumps are price changes or log returns, separated by a random waiting time between trades.

If the jumps 𝑌_𝑖 belong to the domain of attraction of a stable law with index 𝛼, (0<𝛼<2), and the waiting times 𝐽_𝑖 belong to the domain of attraction of a stable law with index 𝛽, (0<𝛽<1), Becker-Kern et al. [11] and Meerschaert and Scheffler [12] showed that as 𝑐→∞, 𝑐^−𝛽/𝛼𝑋([𝑐𝑡])⟹𝐴(𝐸(𝑡))(1.2) a non-Markovian limit with scaling 𝐴(𝐸(𝑐𝑡))=^d𝑐^𝛽/𝛼𝐴(𝐸(𝑡)), where 𝐴(𝑡) is a stable Lévy motion and 𝐸(𝑡) is the inverse or hitting time process of a stable subordinator. Densities of the CTRW scaling limit 𝐴(𝐸(𝑡)) solve a space-time fractional diffusion equation that also involves a fractional time derivative of order 𝛽; see Meerschaert and Scheffler [13], Becker-Kern et al. [11], and Meerschaert and Scheffler [12] for complete details. Becker-Kern et al. [14], Meerschaert and Scheffler [15], and Meerschaert et al. [16] discussed the related limit theorems for CTRWs based on two time scales, triangular arrays and dependent jumps, respectively. The aim of the present paper is to investigate the laws of the iterated logarithm for CTRWs. We establish Chover-type laws of the iterated logarithm for CTRWs with jumps and waiting times in the domains of attraction of stable laws.

Throughout this paper we will use 𝐶 to denote an unspecified positive and finite constant which may be different in each occurrence and use “i.o.” to stand for “infinitely often” and “a.s." to stand for “almost surely” and “𝑢(𝑥)∼𝑣(𝑥)” to stand for “lim𝑢(𝑥)/𝑣(𝑥)=1”. Our main results read as follows.

Theorem 1.1. Let {𝑌_𝑖} be a sequence of i.i.d. nonnegative random variables with a common distribution 𝐹, and let {𝐽_𝑖}, independent of {𝑌_𝑖}, be a sequence of i.i.d. nonnegative random variables with a common distribution 𝐺. Assume that 1−𝐹(𝑥)∼𝑥^−𝛼𝐿(𝑥), 0<𝛼<2, where 𝐿 is a slowly varying function, and that 𝐺 is absolutely continuous and 1−𝐺(𝑥)∼𝐶𝑥^−𝛽, 0<𝛽<1. Let {𝐵(𝑛)} be a sequence such that 𝑛𝐿(𝐵(𝑛))/𝐵(𝑛)^𝛼→𝐶 as 𝑛→∞. Then one has limsup_𝑡→∞𝐵𝑡^𝛽⁻¹𝑋(𝑡)^{1/(loglog𝑡)}=𝑒^1/𝛼a.s.(1.3)

The following is an immediate consequence of Theorem 1.1.

Corollary 1.2. If the tail distribution of 𝑌_𝑖 satisfies 𝑃(𝑌₁>𝑥)∼𝐶𝑥^−𝛼 in Theorem 1.1, then one has limsup_𝑡→∞𝑡^−𝛽/𝛼𝑋(𝑡)^{1/(loglog𝑡)}=𝑒^1/𝛼a.s.(1.4)

In the course of our arguments we often make statements that are valid only for sufficiently large values of some index. When there is no danger of confusion, we omit explicit mention of this proviso.

2. Chung Type LIL for Stable Summands

In this section we consider a Chung-type law of the iterated logarithm for sums of random variables in the domain of attraction of a stable law, which will take a key role to show Theorem 1.1. When 𝐽_𝑖 has a symmetric stable distribution function 𝐺 characterized by 𝐸exp𝑖𝑡𝐽_𝑖=exp−|𝑡|^𝛽for𝑡∈ℝ,(2.1)0<𝛽<2. Chover [17] established that limsup_𝑛→∞||𝑛^−1/𝛽𝑇(𝑛)||^{1/(loglog𝑛)}=𝑒^1/𝛽a.s.(2.2) We call (2.2) as Chover's law of the iterated logarithm. Since then, several papers have been devoted to develop Chover's LIL; see, for example, Hedye [18–20], Pakshirajan and Vasudeva [21], Vasudeva [22], Qi and Cheng [23], Scheffler [24], Chen [25], and Peng and Qi [26] for reference. For some reason the obvious corresponding statement for the “lim  inf” result does not seem to have been recorded, and it is the purpose of this section to do so and may be of independent interest.

Theorem 2.1. Let {𝐽_𝑖} be a sequence of i.i.d. nonnegative random variables with a common distribution 𝐺(𝑥), and let 𝑉(𝑥)=inf{𝑦>0∶1−𝐺(𝑦)≤1/𝑥}. Assume that 𝐺 is absolutely continuous and 1−𝐺(𝑥)∼𝑥^−𝛽𝑙(𝑥), 0<𝛽<1, where 𝑙 is a slowly varying function. Then one has liminf_𝑛→∞𝑉(𝑛)⁻¹𝑇(𝑛)^{1/(loglog𝑛)}=1a.s.(2.3)

In order to prove Theorem 2.1, we need some lemmas.

Lemma 2.2. Let ℎ(𝑥) be a slowly varying function. Then, if 𝑦_𝑛→∞, 𝑧_𝑛→∞, one has for any given 𝜏>0, lim𝑧^𝑛−𝜏ℎ𝑦_𝑛𝑧_𝑛ℎ𝑦_𝑛=0,lim𝑧^𝜏𝑛ℎ𝑦_𝑛𝑧_𝑛ℎ𝑦_𝑛=∞.(2.4)

Proof. See Seneta [27].

Lemma 2.3. Let {𝐽_𝑖} be a sequence of i.i.d. nonnegative random variables with a common distribution 𝐺 and let 𝑀(𝑛)=max{𝐽₁,𝐽₂,…,𝐽_𝑛}. Assume that 𝐺 is absolutely continuous and 1−𝐺(𝑥)∼𝑥^−𝛽𝑙(𝑥), 0<𝛽<1, where 𝑙 is a slowly varying function. Then one has for some given small 𝑡>0lim_𝑛→∞𝐸𝑒^{𝑡𝑇(𝑛)/𝑀(𝑛)}=𝑒^𝑡1−𝑡∫¹⁰𝑒^𝑡𝑥𝑥^−𝛽−1𝑑𝑥.(2.5)

Proof. We will follow the argument of Lemma 2.1 in Darling [28]. Without loss of generality we can assume 𝐽₁=max{𝐽₁,𝐽₂,…,𝐽_𝑛}=𝑀(𝑛) since each 𝐽_𝑖 has a probability of 1/𝑛 of being the largest term, and 𝑃(𝐽_𝑖=𝐽_𝑗)=0 for 𝑖≠𝑗 since 𝐺(𝑥) is presumed continuous.
For notational simplicity we will use the tail distribution 𝐺(𝑥)=1−𝐺(𝑥)=𝑃(𝐽₁>𝑥) and denote by 𝑔(𝑥) the corresponding density, so that 𝐺(𝑥)=∫^∞𝑥𝑔(𝑧)𝑑𝑧. Then, the joint density of 𝐽₁,𝐽₂,…,𝐽_𝑛, given 𝐽₁=𝑀(𝑛), is 𝑔𝑥₁,𝑥₂,…,𝑥_𝑛=𝑛𝑔𝑥¹0𝑔𝑥₂⋯𝑔𝑥_𝑛oifth𝑥e₁rw=ismea.x_𝑖𝑥_𝑖,(2.6) Thus 𝐸𝑒^{𝑡𝑇(𝑛)/𝑀(𝑛)}=⋯𝑒^{𝑡(𝑥1+𝑥2+⋯+𝑥𝑛)/𝑥1}𝑔𝑥₁,𝑥₂,…,𝑥_𝑛𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑛=𝑛𝑒^𝑡^∞0^𝑦0=𝑛𝑒_𝑡⋯^𝑦0𝑒^{𝑡(𝑥2+𝑥3+⋯+𝑥𝑛)/𝑦}𝑔𝑥₂𝑔𝑥₃⋯𝑔𝑥_𝑛𝑔(𝑦)𝑑𝑥₂𝑑𝑥₃⋯𝑑𝑥_𝑛𝑑𝑦^∞0^𝑦0𝑒^{𝑡𝑥/𝑦}𝑔(𝑥)𝑑𝑥^𝑛−1𝑔(𝑦)𝑑𝑦.(2.7) Let us put 𝜙(𝑦,𝑡)=𝑦¹⁰𝑒^𝑡𝑥𝑔(𝑥𝑦)𝑑𝑥(2.8) so that 𝐸𝑒^{𝑡𝑇(𝑛)/𝑀(𝑛)}=𝑛𝑒^𝑡^∞0(𝜙(𝑦,𝑡))^𝑛−1𝑔(𝑦)𝑑𝑦.(2.9) It follows from Doeblin's theorem that if 𝜆>0, 𝐺(𝜆𝑦)=𝜆^−𝛽𝐺(𝑦)(1+𝑜(1))(2.10) for 𝑦≥𝑦₀ with some large 𝑦₀>0. Then, for 𝑦≤𝑦₀, we can choose 𝑡>0 small enough such that 𝑡<−log𝐺(𝑦₀) since 𝐺 has regularly varying tail distribution, so that 𝜙(𝑦,𝑡)≤𝑒^𝑡𝐺𝑦₀<1.(2.11) It follows that 𝑛𝑒^𝑡^𝑦0₀(𝜙(𝑦,𝑡))^𝑛−1𝑔(𝑦)𝑑𝑦⟶0.(2.12) Consider the case 𝑦≥𝑦₀. By a slight transformation we find that 𝜙(𝑦,𝑡)==11−−𝐺(𝑦)+𝑡¹⁰𝑒^𝑡𝑥𝐺(𝑥𝑦)−𝐺(𝑦)𝑑𝑥𝐺(𝑦)+𝑡𝐺(𝑦)(1+𝑜(1))¹⁰𝑒^𝑡𝑥𝑥^−𝛽−1𝑑𝑥.(2.13) Putting 𝜂=𝜂(𝑡)=𝑡¹⁰𝑒^𝑡𝑥𝑥^−𝛽−1𝑑𝑥,(2.14) we have 𝜂<1 since 0<𝛽<1 and 𝑡 is small. Thus 𝜙(𝑦,𝑡)=1−𝐺(𝑦)(1−𝜂)+𝑜𝐺(𝑦).(2.15) By (2.9) and making the change of variable 𝑛𝐺(𝑦)=𝑣 to give 𝐸𝑒^{𝑡𝑇(𝑛)/𝑀(𝑛)}=𝑒^𝑡^𝑛01−𝑣𝑛(1−𝜂)+𝑣𝑜1𝑛^𝑛−1𝑑𝑣⟶𝑒^𝑡^∞0=𝑒𝑒^{−𝑣(1−𝜂)}𝑑𝑣_𝑡1−𝜂,(2.16) which yields the desired result.

The following large deviation result for stable summands is due to Heyde [19].

Lemma 2.4. Let {𝜉_𝑖} be a sequence of i.i.d. nonnegative random variables with a common tail distribution satisfying 𝑃(𝜉₁>𝑥)∼𝑥^−𝑟ℎ(𝑥), 0<𝑟<2, where ℎ is a slowly varying function. Let {𝜆_𝑛} be a sequence such that 𝑛ℎ(𝜆_𝑛)/𝜆^𝑟𝑛→𝐶 as 𝑛→∞, and let {𝑥_𝑛} be a sequence with 𝑥_𝑛→∞ as 𝑛→∞. Then 0<liminf_𝑛→∞𝑥^𝑟𝑛ℎ𝜆_𝑛ℎ𝑥_𝑛𝜆_𝑛𝑃^𝑛_𝑖=1𝜉_𝑖>𝑥_𝑛𝜆_𝑛≤limsup_𝑛→∞𝑥^𝑟𝑛ℎ𝜆_𝑛ℎ𝑥_𝑛𝜆_𝑛𝑃^𝑛_𝑖=1𝜉_𝑖>𝑥_𝑛𝜆_𝑛<∞.(2.17)

Now we can show Theorem 2.1.

Proof of Theorem 2.1. In order to show (2.3), it is enough to show that for all 𝜀>0lliimmiinnf_𝑛→∞f_𝑛→∞(log𝑛)^𝜀𝑉(𝑛)⁻¹𝑇(𝑛)≥1a.s.,(2.18)(log𝑛)^−𝜀𝑉(𝑛)⁻¹𝑇(𝑛)≤1a.s.(2.19)
We first show (2.18). Let 𝑛_𝑘=[𝜃^𝑘], 1<𝜃<2. Put again 𝐺(𝑥)=1−𝐺(𝑥)=𝑃(𝐽₁>𝑥). Let 𝐺^∗ be the inverse of 𝐺. Obverse that 𝐺^∗(𝑦)∼𝑦^−1/𝛽𝐻(1/𝑦), 0<𝑦≤1, where 𝐻 is a slowly varying function and 𝑉(𝑛)=𝐺^∗(1/𝑛)∼𝑛^1/𝛽𝐻(𝑛), so that 𝑉𝑛_𝑘𝑉𝑛_𝑘+1⟶𝜃^−1/𝛽log𝑛_𝑘(2.20)^−𝜀𝑉𝑛_𝑘𝐺^∗log𝑛^𝑘^𝛽𝜀/2𝑛^𝑘−1∼log𝑛^𝑘^−𝜀𝑛^𝑘1/𝛽𝐻𝑛_𝑘=log𝑛_𝑘𝑛^𝑘1/𝛽(log𝑛)^−𝜀/2𝐻𝑛_𝑘log𝑛_𝑘^{−𝛽𝜀/2}^−𝜀/2𝐻𝑛_𝑘𝐻𝑛_𝑘log𝑛_𝑘^{−𝛽𝜀/2}⟶0,(2.21) by Lemma 2.2. Let 𝑈,𝑈₁,𝑈₂,…,𝑈_𝑛 be i.i.d. random variables with the distribution of 𝑈 Uniform over (0,1), and let 𝑀^∗(𝑛)=max{𝑈₁,𝑈₂,…,𝑈_𝑛}. Then, from the fact that 𝐺(𝐽_𝑛) is a Uniform (0,1) random variable, we note that 𝑀^∗(𝑛)=^d𝐺(𝑀(𝑛)), 𝑛≥1. From (2.21), 𝐽_𝑖 nonnegative, and 𝐺 and 𝐺^∗ nonincreasing, it follows that 𝑃𝑇≤𝑛𝑃_𝑘𝑀≤l𝑛o_𝑘g𝑛_𝑘^−𝜀𝑉𝑛_𝑘≤𝑃(≤log𝑛_𝑘^−𝜀𝑉𝑛_𝑘=𝑃(𝐺^∗𝐺𝑀𝑛_𝑘≤𝐺^∗log𝑛^𝑘^𝛽𝜀/2𝑛^𝑘−1=𝑃𝐺1−𝑀𝑀_∗𝑛_𝑘≥log𝑛_𝑘^𝛽𝜀/2𝑛^𝑘−1=𝑃𝑀_∗𝑛_𝑘≥log𝑛_𝑘^𝛽𝜀/2𝑛^𝑘−1=𝑃𝑈≤𝑛1^𝑘−≤l1og−𝑛_𝑘log𝑛_𝑘^𝛽𝜀/2𝑛^𝑘−1^𝛽𝜀/2𝑛^𝑘−1^𝑛𝑘≤exp−log𝑛_𝑘^𝛽𝜀/2.(2.22) Hence, the sum of the left hand side of the previously mentioned probability is finite; by the Borel-Cantelli lemma, we get liminf_𝑘→∞log𝑛_𝑘^𝜀𝑉𝑛_𝑘⁻¹𝑇𝑛_𝑘≥1a.s.(2.23) Thus, by (2.20) we have liminf_𝑛→∞(≥logli𝑛m)^𝜀i𝑉nf_𝑘→∞(𝑛)⁻¹𝑇(𝑛)≥liminf_𝑘→∞min_{𝑛𝑘≤𝑛≤𝑛𝑘+1}(log𝑛)^𝜀𝑉(𝑛)⁻¹𝑇(𝑛)𝑉𝑛^𝑘𝑉𝑛_𝑘+1log𝑛^𝑘≥𝜃_−1/𝛽^𝜀𝑉𝑛_𝑘⁻¹𝑇𝑛_𝑘a.s.(2.24) Therefore, by the arbitrariness of 𝜃>1, (2.18) holds.
We now show (2.19). Let 𝑛_𝑘=[𝑒^𝑘1+𝛿], 𝛿>0. For notational simplicity, we introduce the following notations: 𝜁_𝑘=𝑇𝑛^𝑘−𝑛_𝑘−1𝐸_𝑘𝑀𝑛_𝑘−𝑛_𝑘−1,=𝐸𝑇𝑛_𝑘−𝑇𝑛_𝑘−1≤log𝑛_𝑘^𝜀𝑉𝑛_𝑘_𝑘,𝐹=𝑇𝑛_𝑘−1≥𝜀log𝑛_𝑘^𝜀𝑉𝑛_𝑘_𝑘,=𝑂_𝑘𝑀𝑛_𝑘−𝑛_𝑘−1≤loglog𝑛_𝑘^{(1−𝜀)/𝛽}𝑉𝑛_𝑘,=𝜁_𝑘≥log𝑛_𝑘^𝜀loglog𝑛_𝑘^{−(1−𝜀)/𝛽}.(2.25) By Lemma 2.3, we have 𝑃𝑂_𝑘≤exp−𝑡log𝑛_𝑘^𝜀loglog𝑛_𝑘^{−1−𝜀/𝛽}𝐸𝑒^𝑡𝜁𝑘≤𝐶exp−𝑡log𝑛_𝑘^𝜀loglog𝑛_𝑘^{−(1−𝜀)/𝛽}.(2.26) Thus, we get ∑𝑃(𝑂_𝑘)<∞.
Observe again that 𝐺^∗(𝑦)∼𝑦^−1/𝛽𝐻(1/𝑦) and 𝑉(𝑛)∼𝑛^1/𝛽𝐻(𝑛), so that 𝑉𝑛_𝑘𝑉𝑛_𝑘−1≥𝑒^{(1/𝛽)𝑘𝛿}loglog𝑛_𝑘,(2.27)^{(1−𝜀)/𝛽}𝑉𝑛_𝑘𝐺^∗loglog𝑛^𝑘^(1−𝜀)𝑛^𝑘−1∼loglog𝑛^𝑘^{2(1−𝜀)/𝛽}𝐻𝑛_𝑘𝐻loglog𝑛_𝑘^{−(1−𝜀)}𝑛_𝑘⟶∞,(2.28) by Lemma 2.2. Thus, we note 𝑃𝐹_𝑘≥=𝑃𝑃((𝐺^∗𝐺𝑀𝑛_𝑘−𝑛_𝑘−1≤𝐺^∗loglog𝑛^𝑘^(1−𝜀)𝑛^𝑘−1=𝑃𝐺1−𝑀𝑀_∗𝑛_𝑘−𝑛_𝑘−1≥loglog𝑛_𝑘^(1−𝜀)𝑛^𝑘−1=𝑃𝑀_∗𝑛_𝑘−𝑛_𝑘−1≥loglog𝑛_𝑘^(1−𝜀)𝑛^𝑘−1=𝑃𝑈≤𝑛1^𝑘−−𝑛log^𝑘−1lo≤g1𝑛_𝑘−loglog𝑛_𝑘^(1−𝜀)𝑛^𝑘−1^(1−𝜀)𝑛^𝑘−1^{𝑛𝑘−𝑛𝑘−1}=1−loglog𝑛_𝑘^(1−𝜀)𝑛^𝑘−1^{𝑛𝑘−𝑛𝑘−1}≥exp−𝐶loglog𝑛_𝑘^(1−𝜀/2),(2.29) which yields easily ∑𝑃(𝐹_𝑘)=∞. Hence, since 𝑃(𝐸_𝑘)≥𝑃(𝐹_𝑘)−𝑃(𝑂_𝑘), we get ∑𝑃(𝐸_𝑘)=∞. Since 𝐸_𝑘 are independent, by the Borel-Cantelli lemma, we get liminf_𝑘→∞log𝑛_𝑘^−𝜀𝑉𝑛_𝑘⁻¹𝑇𝑛_𝑘−𝑇𝑛_𝑘−1≤1a.s.(2.30)
By applying Lemma 2.4 and (2.27) and some simple calculation, we have easily that ∑𝑃(𝐸_𝑘)<∞, so that limsup_𝑘→∞log𝑛_𝑘^−𝜀𝑉𝑛_𝑘⁻¹𝑇𝑛_𝑘−1=0a.s.,(2.31) which, together with (2.30), implies liminf_𝑘→∞log𝑛_𝑘^−𝜀𝑉𝑛_𝑘⁻¹𝑇𝑛_𝑘≤1a.s.(2.32) This yields (2.19). The proof of Theorem 2.1 is now completed.

3. Proof of Theorem 1.1

Proof of Theorem 1.1. We have to show that for all 𝜀>0lliimmssuup_𝑡→∞p_𝑡→∞(log𝑡)^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽⁻¹𝑋(𝑡)≤1a.s.,(3.1)(log𝑡)^{−(1−𝜀)/𝛼}𝐵𝑡^𝛽⁻¹𝑋(𝑡)≥1a.s.(3.2)
We first show (3.1). Let 𝑡_𝑘=𝜃^𝑘, 1<𝜃<2. For notational simplicity, we introduce the following notations: 𝑈(𝑥)=𝑄(_𝑘lo=g𝑥)_−𝜌log𝑡_𝑘^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑆𝑁_𝑡𝑘≥1,𝑥^1/𝛽,𝛾₁(𝑥)=sup{𝑦∶𝑈(𝑦)≤𝑥},𝜌=𝜀𝑄5𝛽_𝑘,=log𝑡_𝑘𝑅^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑆𝛾₁𝑡_𝑘≥1_𝑘,=𝑁_𝑡𝑘≥𝛾₁𝑡_𝑘.(3.3)
By (2.18), we have 𝑃𝑅_𝑘i.o.=𝑃𝑇𝛾₁𝑡_𝑘≤𝑡_𝑘i.o.=𝑃𝑇𝑡_𝑘≤log𝑡_𝑘^−𝜌𝑉𝑡_𝑘i.o.=0.(3.4)
Put 𝐹(𝑥)=1−𝐹(𝑥)=𝑃(𝑌₁>𝑥). Let 𝐹^∗ be the inverse of 𝐹. Recall that 𝐹^∗(𝑦)∼𝑦^−1/𝛼𝐻(1/𝑦), 0<𝑦≤1, where 𝐻 is a slowly varying function, so that 𝐵(𝑛)=𝐹^∗(𝐶/𝑛)∼𝐶𝑛^1/𝛼𝐻(𝑛) and 𝐵𝑡^𝛽𝑘𝐵𝑡^𝛽𝑘−1⟶𝜃^𝛽/𝛼.(3.5)
Note that 𝑈log𝑡_𝑘^𝜀/4𝑡^𝛽𝑘∼log𝑡^𝑘^𝜀/(4𝛽)𝑡_𝑘loglog𝑡^𝑘^𝜀/4𝑡^𝛽𝑘^−𝜌≥𝑈𝛾₁𝑡_𝑘=𝑡_𝑘.(3.6) Thus, by noting 𝑈 increasing, log𝑡_𝑘^𝜀/(4𝛼)𝑡^{𝑘𝛽/𝛼}≥𝛾₁𝑡_𝑘^1/𝛼.(3.7) Hence, by Lemma 2.2, log𝑡_𝑘^𝜀/(2𝛼)𝐵𝑡^𝛽𝑘𝐵𝛾₁𝑡_𝑘≥𝐶log𝑡^𝑘^𝜀/(2𝛼)𝑡^{𝑘𝛽/𝛼}𝐻𝑡^𝛽𝑘^1/𝛼𝛾₁𝑡_𝑘^1/𝛼𝐻𝛾¹𝑡_𝑘^1/𝛼≥1.(3.8) Thus, by (3.8) and Lemma 2.4, we have 𝑃𝑄_𝑘≤𝑃⎛⎜⎜⎝𝑆𝛾¹𝑡_𝑘≥⎛⎜⎜⎝log𝑡_𝑘^{(1+𝜀)/𝛼}𝐵𝑡^𝛽𝑘𝐵𝛾₁𝑡_𝑘⎞⎟⎟⎠𝐵𝛾¹≤𝑃𝑆𝛾₁𝑡_𝑘⎞⎟⎟⎠≤𝐶log𝑡_𝑘𝑡_𝑘≥log𝑡_𝑘^{(1+𝜀/2)/𝛼}𝐵𝛾₁𝑡_𝑘^{−(1+𝜀/4)}.(3.9) Therefore, ∑𝑃(𝑄_𝑘)<∞. By the Borel-Cantelli lemma, we get 𝑃(𝑄_𝑘i.o.)=0.
Observe that 𝑃^∞_𝑘=𝑛𝑄_𝑘=𝑃^∞_𝑘=𝑛𝑄_𝑘∩^∞_𝑘=𝑛𝑅^𝑐𝑘+𝑃^∞_𝑘=𝑛𝑄_𝑘∩^∞_𝑘=𝑛𝑅^𝑐𝑘≤𝑃_∞^𝑐_𝑘=𝑛𝑄_𝑘+𝑃^∞_𝑘=𝑛𝑅_𝑘,(3.10) where 𝐸^𝑐 stands for the complement of 𝐸. Thus, letting 𝑛→∞, we have 𝑃𝑄_𝑘i.o.≤𝑃𝑄_𝑘i.o.+𝑃𝑅_𝑘i.o.=0,(3.11) which implies that limsup_𝑘→∞log𝑡_𝑘^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑋𝑡_𝑘≤1a.s.(3.12) Thus, by (3.5), we have limsup_𝑡→∞(≤logli𝑡m)sup^{−(1+𝜀)/𝛼}_𝑘→∞𝐵𝑡^𝛽⁻¹𝑋(𝑡)≤𝜃_𝛽/𝛼max_{𝑡𝑘−1<𝑡≤𝑡𝑘}(log𝑡)^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽⁻¹𝑋(𝑡)≤𝜃_𝛽/𝛼limsup_𝑘→∞log𝑡_𝑘^{−(1+𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑋𝑡_𝑘a.s.(3.13) This yields (3.1) immediately by letting 𝜃↓1.
We now show (3.2). Let 𝑡_𝑘=𝑒^𝑘1+𝛿, 𝛿>0. To show (3.2), it is enough to prove limsup_𝑘→∞log𝑡_𝑘^{−(1−𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑋𝑡_𝑘≥1a.s.(3.14)
Put 𝑈Λ_𝑘=log𝑡_𝑘^{−(1−𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑆𝑁_𝑡𝑘≥1₁,(𝑥)=(log𝑥)^𝜌𝑥^1/𝛽,𝛾₂(𝑥)=sup𝑦∶𝑈₁(𝑦)≤𝑥,𝜌=𝜀𝑊5𝛽_𝑘,=log𝑡_𝑘^{−(1−𝜀)/𝛼}𝐵𝑅𝑡^𝛽𝑘⁻¹𝑆𝛾₂𝑡_𝑘−𝑆𝛾₂𝑡_𝑘−1≥1_𝑘,=𝑁_𝑡𝑘≥𝛾₂𝑡_𝑘.(3.15)
By (2.19), we have 𝑃𝑅_𝑘i.o.=𝑃𝑇𝛾₂𝑡_𝑘≤𝑡_𝑘i.o.=𝑃𝑇𝑡_𝑘≤log𝑡_𝑘^𝜀𝑡^𝑘1/𝛽i.o.=1.(3.16)
Note that 𝑈₁log𝑡^𝑘^−𝜀/4𝑡^𝛽𝑘∼log𝑡^𝑘^{−𝜀/(4𝛽)}𝑡_𝑘loglog𝑡^𝑘^−𝜀/4𝑡^𝛽𝑘^𝜌≤𝑈₁𝛾₂𝑡_𝑘=𝑡_𝑘.(3.17) Thus, by noting 𝑈₁ increasing, log𝑡_𝑘^{−𝜀/(4𝛼)}𝑡^{𝑘𝛽/𝛼}≤𝛾₂𝑡_𝑘^1/𝛼.(3.18) Hence, by Lemma 2.2, log𝑡_𝑘^{−𝜀/(2𝛼)}𝐵𝑡^𝛽𝑘𝐵𝛾₂𝑡_𝑘≤𝐶log𝑡^𝑘^{−𝜀/(2𝛼)}𝑡^{𝑘𝛽/𝛼}𝐻𝑡^𝛽𝑘^1/𝛼𝛾₂𝑡_𝑘^1/𝛼𝐻𝛾²𝑡_𝑘^1/𝛼⟶0.(3.19) Similarly, by noting 𝑡_𝑘/𝑡_𝑘−1→∞, one can have 𝐵𝛾₂𝑡_𝑘𝐵𝛾₂𝑡_𝑘−𝛾₂𝑡_𝑘−1⟶1.(3.20) Thus, by Lemma 2.4, we have 𝑃𝑊_𝑘≥𝑃⎛⎜⎜⎝𝑆𝛾₂𝑡_𝑘−𝛾₂𝑡_𝑘−1≥⎛⎜⎜⎝log𝑡_𝑘^{(1−𝜀)/𝛼}𝐵𝑡^𝛽𝑘𝐵𝛾₂𝑡_𝑘⎞⎟⎟⎠𝐵𝛾²≥𝑃𝑆𝛾₂𝑡_𝑘⎞⎟⎟⎠≥𝐶log𝑡_𝑘𝑡_𝑘−𝛾₂𝑡_𝑘−1≥log𝑡_𝑘^{(1−𝜀/2)/𝛼}𝐵𝛾₂𝑡_𝑘^{−(1−𝜀/4)}.(3.21) Therefore, ∑𝑃(𝑊_𝑘)=∞. Since the events {𝑊_𝑘} are independent, by the Borel-Cantelli lemma, we get 𝑃(𝑊_𝑘i.o.)=1.
Now, observe that 𝑃^∞_𝑛=𝑚Λ_𝑘≥𝑃^∞_𝑛=𝑚Λ^𝑘≥𝑃_∞∩𝑅_𝑘_𝑛=𝑚log𝑡^𝑘^{−(1−𝜀)/𝛼}𝐵𝑡^𝛽𝑘⁻¹𝑆𝛾₂𝑡_𝑘≥1×𝑃^∞_𝑛=𝑚𝑅_𝑘≥𝑃_∞_𝑛=𝑚𝑊_𝑘×𝑃^∞_𝑛=𝑚𝑅_𝑘.(3.22) Therefore, by letting 𝑚→∞, we get 𝑃Λ_𝑘i.o.≥𝑃𝑊_𝑘i.o.−𝑃𝑊_𝑘i.o.𝑃𝑅_𝑘i.o.=1,(3.23) which implies (3.14). The proof of Theorem 1.1 is now completed.

Remark 3.1. By the proof Theorem 1.1, (1.3) can be modified as follows: limsup_𝑡→∞(log𝑡)^−1/𝛼𝐵𝑡^𝛽⁻¹𝑋(𝑡)=1a.s.(3.24) That is to say that the form of (1.3) is no rare and the variables (𝐵(𝑡^𝛽))⁻¹𝑋(𝑡) must be cut down additionally by the factors (log𝑡)^−1/𝛼 to achieve a finite lim  sup.

Acknowledgments

The authors wish to express their deep gratitude to a referee for his/her valuable comments on an earlier version which improve the quality of this paper. K. S. Hwang is supported by the Korea Research Foundation Grant Funded by Korea Government (MOEHRD) (KRF-2006-353-C00004), and W. Wang is supported by NSFC Grant 11071076.