PROBABILITY
AND
MATHEMATICAL STATISTICS
Vol. 37, Fasc. 1 (2017), pp. 185–199
doi:10.19195/0208-4147.37.1.8
STRONG LAW OF LARGE NUMBERS FOR RANDOM VARIABLES
WITH MULTIDIMENSIONAL INDICES
BY
AGNIESZKA M. G D U L A (LUBLIN) AND ANDRZEJ K R A J K A
∗
(LUBLIN)
Abstract. Let {Xn ,n ∈ V ⊂ N
2
} be a two-dimensional random field
of independent identically distributed random variables indexed by some
subset V of lattice N
2
. For some sets V the strong law of large numbers
lim
n →∞,n ∈V
∑
k ∈V,k n
X
k
|n |
= µ a.s.
is equivalent to
EX1 = µ and
∑
n ∈V
P [|X1 | > |n |] < ∞.
In this paper we characterize such sets V .
2010 AMS Mathematics Subject Classification: Primary: 60F15;
Secondary: 60G50, 60G60.
Key words and phrases: Strong law of large numbers, sums of ran-
dom fields, multidimensional index.
1. INTRODUCTION
Let {X
n
,n =(n
1
,n
2
,...,n
d
) ∈ N
d
} be a family of independent identically
distributed random variables indexed by N
d
-vectors, and let us put
S
n
=
∑
k n
X
k
, n ∈ N
d
,
where k n iff k
j
n
j
,j =1, 2,...,d. In this paper we investigate the almost
sure behavior of the sums S
n
when |n |
def
=
∏
d
j =1
n
j
→∞, i.e., the strong law of
large numbers (SLLN).
∗
Corresponding author.
Probability and Mathematical Statistics 37, z. 1, 2017
© for this edition by CNS