Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

Abstract

Let {𝑋,𝑋_𝑛;𝑛∈𝑍^𝑑₊} be a sequence of i.i.d. real-valued random variables, and 𝑆_𝑛=∑_𝑘≤𝑛𝑋_𝑘, 𝑛∈𝑍^𝑑₊. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series ∑_𝑛𝑏(𝑛)𝜓²(𝑎(𝑛))𝑃{|𝑆_𝑛|≥𝑎(𝑛)𝜙(𝑎(𝑛))}, where 𝑎(𝑛)=𝑛^1/𝛼1₁⋯𝑛^1/𝛼𝑑_𝑑, 𝑏(𝑛)=𝑛^𝛽1₁⋯𝑛^𝛽𝑑_𝑑, 𝜙 and 𝜓 are taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).

1. Introduction

Let 𝑍^𝑑₊ be the set of all positive integer 𝑑-dimensional lattice points with coordinate-wise partial ordering, ≤; that is, for every 𝑚=(𝑚₁,…,𝑚_𝑑), 𝑛=(𝑛₁,…,𝑛_𝑑)∈𝑍^𝑑₊, 𝑚≤𝑛 if and only if 𝑚_𝑖≤𝑛_𝑖, 𝑖=1,2,…,𝑑, where 𝑑≥1 is a fixed integer. |𝑛| denote ∏^𝑑_𝑖=1𝑛_𝑖 and 𝑛→∞ means 𝑛_𝑖→∞,𝑖=1,2,…,𝑑. Throughout the paper, {𝑋,𝑋_𝑛,𝑋_𝑛;𝑛∈𝑍₊,𝑛∈𝑍^𝑑₊} are i.i.d. random variables with 𝐸𝑋=0 and 𝐸𝑋²=𝜎². Let 𝑆_𝑛=∑^𝑛_𝑘=1𝑋_𝑘, and 𝑆_𝑛=∑_𝑘≤𝑛𝑋_𝑘. we define

𝑎(𝑛)=𝑛^1/𝛼1₁⋯𝑛^1/𝛼𝑑_𝑑,𝑏(𝑛)=𝑛^𝛽1₁⋯𝑛^𝛽𝑑_𝑑, (1.1) where 𝛼₁,…,𝛼_𝑑;𝛽₁,…,𝛽_𝑑 are real numbers with 0<𝛼_𝑖≤2,𝛽_𝑖≥−1,𝑖=1,2,…,𝑑. Further, set 𝛼=max{𝛼_𝑖;1≤𝑖≤𝑑},𝑠=max{𝛼_𝑖(𝛽_𝑖+1);1≤𝑖≤𝑑},𝑟=max{𝛼_𝑖(𝛽_𝑖𝛼+2);1≤𝑖≤𝑑},𝑞=card{𝑖∶_𝑖(𝛽_𝑖+2)=𝑟}, and 𝑝=card{𝑖∶𝛼_𝑖(𝛽_𝑖+2)=1}.

Hartman and Wintner [1] studied the fundamental strong laws of classic Probability Theory for i.i.d random variables {𝑋,𝑋_𝑛;𝑛∈𝑍₊} and obtained the following Hartman-Wintner law of the iterated logarithm (LIL).

Theorem 1.1. 𝐸𝑋=0 and 𝐸𝑋²=𝜎²<∞ if and only if limsup_𝑛→∞𝑆_𝑛(2𝑛loglog𝑛)^1/2=−liminf_𝑛→∞𝑆_𝑛(2𝑛loglog𝑛)^1/2=𝜎𝑎.𝑠. (1.2)

Afterward, the study of the estimate of the convergence rate in the above relation (1.2) has been attracting the attention of various researchers over the last few decades. Darling and Robbins [2], Davis [3], Gafurov [4], and Li [5] have obtained some good results on the estimate of convergence rate in (1.2). The best result is probably the one given by Li et al. [6]. For easy reference, we restate their result in Theorem 1.2.

Theorem 1.2. Let 𝜙(𝑡) and 𝜓(t) be two positive real-valued functions on [1,∞) such that 𝜙(t) is nondecreasing, lim_t→∞𝜙(t)=∞, and 𝜓(t)=O(𝜙(t)) as t→∞. For t≥0, let 𝜎²(t)=EX²I{|X|<√t}−(𝐸𝑋𝐼{|𝑋|<√𝑡})², and 𝜎=²_𝑛^Δ𝜎²(𝑛𝜙²(𝑛)). Then the following results are equivalent: _𝑛≥1𝜓²(𝑛)𝑛𝑃{|𝑆_𝑛|≥𝑛^1/2𝜙(𝑛)}<∞,_𝑛≥1𝜓²(𝑛)𝜙𝑛𝜙(𝑛)exp(−²(𝑛)2𝜎²_𝑛)<∞. (1.3)(1.4) where 𝐻(𝑥) is the reverse function of 𝑥𝜙(𝑥).

Proof. Because 𝛼_𝑖(𝛽_𝑖+2)≥𝛼_𝑖>0, by using Lemma 3.1, we have _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))|𝑛|𝑃{|𝑋|≥𝑎(𝑛)𝜙(𝑎(≃𝑛))}_∞𝑖=1(_{𝑖≤𝑎(𝑛)<𝑖+1}𝑏(𝑛)𝜓(𝑎(𝑛))|=𝑛|)𝑃{|𝑋|≥𝑖𝜙(𝑖)}_∞𝑖=1(_{𝑖≤𝑎(𝑛)<𝑖+1}𝑏(𝑛)𝜓(𝑎(𝑛))|𝑛|)^∞_𝑗=1=𝑃{𝑗𝜙(𝑗)≤|𝑋|<(𝑗+1)𝜙(𝑗+1)}_∞𝑗=1(_{1≤𝑎(𝑛)<𝑗+1}𝑏(𝑛)𝜓(𝑎(𝑛))|≃𝑛|)𝑃{𝑗≤𝐻(|𝑋|)<(𝑗+1)}_∞𝑗=1𝜓(𝑗+1)(𝑗+1)^𝑟(log(𝑗+1))^{𝑞−1+𝑝}𝑃{𝑗≤𝐻(|𝑋|)<(𝑗+1)}. (4.17) The last expression is finite if and only if 𝐸(𝐻(|𝑋|))^𝑟(log⁺𝐻(|𝑋|))^{𝑞−1+𝑝}𝜓(𝐻(|𝑋|))<+∞. This completes the proof of the lemma.

Lemma 4.2. ∑_𝑛𝑏(𝑛)𝜓(𝑎(𝑛))𝑃{|𝑆_𝑛|≥𝑎(𝑛)𝜙(𝑎(𝑛))}<∞ implies (4.15).

Proof. We first prove the result for symmetric random variables. One may rearrange {𝑋_𝑘,𝑘≤𝑛} and obtain {𝑋_𝑗,1≤𝑗≤|𝑛|}. By Levy inequality, we set 𝑃{max_𝑘≤𝑛|𝑋_𝑘|≥𝑎(𝑛)𝜙(𝑎(𝑛))}=𝑃{max_{1≤𝑗≤|𝑛|}|𝑋_𝑗|≥𝑎(𝑛)𝜙(𝑎(𝑛))}≤2𝑃{|^|𝑛|_𝑗=1𝑋_𝑗|≥𝑎(𝑛)𝜙(𝑎(𝑛))}=2𝑃{|𝑆_𝑛|≥𝑎(𝑛)𝜙(𝑎(𝑛))}<∞, (4.18) but 𝑃{max_𝑘≤𝑛|𝑋_𝑘|≥𝑎(𝑛)𝜙(𝑎(𝑛))}=1−(1−𝑃{|𝑋|≥𝑎(𝑛)𝜙(𝑎(𝑛))})^|𝑛|≈|𝑛|𝑃{|𝑋|≥𝑎(𝑛)𝜙(𝑎(𝑛))}. (4.19) Thus for large |𝑛|,|𝑛|𝑃{|𝑋|≥𝑎(𝑛)𝜙(𝑎𝑛)}≤𝐶𝑃{|𝑆_𝑛|≥𝑎(𝑛)𝜙(𝑎(𝑛))}, (4.20) equation (4.15) follows from Lemma 4.1.
If {𝑋_𝑛,𝑛∈𝑍^𝑑₊} are nonsymmetric random variables, then by using the standard symmetrization method, it is easy to prove that for some constant 𝐶,|𝑛|𝑃{|𝑋|≥𝐶𝑎(𝑛)𝜙(𝑎𝑛)}≤𝐶𝑃{|𝑆_𝑛|≥𝑎(𝑛)𝜙(𝑎(𝑛))}, (4.21) which implies that _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))|𝑛|𝑃{|𝑋|≥𝐶𝑎(𝑛)𝜙(𝑎(𝑛))}<∞. (4.22) That is, (4.15) holds.

Proof of Theorem 2.5. Similar to the proof of Theorem 2.2, by Lemma 4.1, 𝐸(𝐻(|𝑋|))^𝑟(log⁺𝐻(|𝑋|))^{𝑞−1+𝑝}𝜓(𝐻(|𝑋|))<∞ implies that _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))|𝑃{𝑆_𝑛≥𝑎(𝑛)𝜙(𝑎(𝑛))}−Φ(−𝑎(𝑛)𝜙(𝑎(𝑛))|𝑛|^1/2𝜎_𝑎(𝑛))|<∞. (4.23) Therefore, if 𝐸(𝐻(|𝑋|))^𝑟(log⁺𝐻(|𝑋|))^{𝑞−1+𝑝}𝜓(𝐻(|𝑋|))<∞, (2.12) is equivalent to _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))Φ(−𝑎(𝑛)𝜙(𝑎(𝑛))|𝑛|^1/2𝜎_𝑎(𝑛))<∞, (4.24) which is equivalent to _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))|𝑛|^1/2𝑎(𝑛)𝜙(𝑎(𝑎𝑛))exp(−²(𝑛)𝜙²(𝑎(𝑛))2|𝑛|𝜎²_𝑎(𝑛))<∞. (4.25)
On the other hand, (2.12) implies 𝐸(𝐻(|𝑋|))^𝑟(log⁺𝐻(|𝑋|))^{𝑞−1+𝑝}𝜓(𝐻(|𝑋|))<∞ by Lemma 4.2. Hence, (2.11) and (2.12) are equivalent. If, in addition, 𝐸𝑋²𝐼{|𝑋|>𝑎(𝑛)𝜙(𝑎(𝑛))}=𝑂(|𝑛|/𝑎²(𝑛)𝜙²(𝑎(𝑛))), (4.24) is equivalent to the following: _𝑛𝑏(𝑛)𝜓(𝑎(𝑛))|𝑛|^1/2𝑎(𝑛)𝜙(𝑎(𝑎𝑛))exp(−²(𝑛)𝜙²(𝑎(𝑛))2|𝑛|)<∞. (4.26) Hence, (2.12) and (2.13) are equivalent.

Proof of Theorem 2.7. Like the proof of Theorem 2.1, 𝐸𝑋²(log⁺|𝑋|)^𝑑−1<+∞ and (2.17) imply (2.16). On the other hand, if (2.16) holds, then 𝐸𝑋²(log⁺|𝑋|)^𝛿<∞ implies (2.17). The proof is complete.

Acknowledgments

The author is very grateful for referee's comments and suggestions which greatly improve the readability and quality of the current paper. The research is partly supported by the Natural Sciences and Engineering Research Council of Canada.