Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

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Journal of Multivariate Analysis
Volume 86, Issue 2, August 2003, Pages 398–422

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

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  Allan Gut^{a,  ,}  , 
  
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  Aurel Spătaru^b, 
  

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  ^a Department of Mathematics, Uppsala University, P.O. Box 480, SE-75106 Uppsala, Sweden
  
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  ^b Centre of Mathematical Statistics, Romanian Academy, Calea 13 Septembrie No. 13, Ro-761 00 Bucharest 5, Romania
  

• 
  http://dx.doi.org/10.1016/S0047-259X(03)00050-2, How to Cite or Link Using DOI
  

• 
  Permissions & Reprints
  

Abstract

Consider  —the positive d-dimensional lattice points with partial ordering ⩽, let   be i.i.d. random variables with mean 0, and set  ,  . We establish precise asymptotics for  , and for  , (0⩽δ⩽1) as ε↘0, and for   as  .

MSC

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  primary 60G50; 
  
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  60E07; 
  
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  60F10; 
  
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  secondary 60E15
  

Keywords

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  Multidimensional indices; 
  
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  Tail probabilities of sums of i.i.d. random variables; 
  
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  Stable distributions; 
  
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  Domain of attraction; 
  
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  Strong law; 
  
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  Law of the iterated logarithm
  

1. Introduction and results

Let X,X₁,X₂,…, be a sequence of i.i.d. random variables, F is the distribution of X, and Sn=X₁+⋯+Xn. Our point of departure is the result that, for p<2 and r⩾p,
equation(1.1)

if and only if E|X|r<∞, and, when r⩾1, EX=0. For r=2,p=1 the sufficiency was proved by Hsu and Robbins [14], and the necessity by Erdős [3] and [4]. For the case r=p=1 we refer to Spitzer [29], and for the general result to Katz [17] and Baum and Katz [1].
The sums obviously tend to infinity as ε↘0. An interesting problem is to find the exact rate at which this occurs. A first result toward this end was given in Heyde [13], who proved that
equation(1.2)

whenever EX=0 and EX²<∞. Chen [2] proved an analogous result related to the series in (1.1), under the assumption of at least finite variance.
Based on Spătaru [28] (the case p=1 below), Gut and Spătaru [10] proved that, if EX=0, and F belongs to the domain of attraction of a nondegenerate stable distribution G with characteristic exponent α, 1<α⩽2, then, for 1⩽p<α,
equation(1.3)

In the same paper it is also shown that, if EX=0, and F belongs to the normal domain of attraction of a nondegenerate stable distribution G with characteristic exponent α, 1<α⩽2, then, for 1⩽p<r<α,
equation(1.4)

where Z is a random variable having the distribution G. Some analogs related to the law of the iterated logarithm are proved in [11].
Now, let   denote the positive integer d-dimensional lattice with coordinate-wise partial ordering ⩽. The notation  , where   and  , thus means thatmk⩽nk, for k=1,2,…,d. We also use   for ∏k₌₁dnk, and   is to be interpreted as nk→∞, for k=1,2,…,d. Finally, following Gut [7], we set π(j)=(j,1,1,…,1), j⩾1. Throughout the remainder of this paper we assume that X and   are i.i.d. random variables, and set  .
Following is the multiindex analog of (1.1) given in [7].