The Marcinkiewicz–Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences
This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables { X , X n , n ≥ 1 } with general normalizing constants under a moment condition that E R (...
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| Published in: | Journal of theoretical probability Vol. 34; no. 1; pp. 331 - 348 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
Springer US
01-03-2021
Springer Nature B.V |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | This paper establishes complete convergence for weighted sums and the Marcinkiewicz–Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables
{
X
,
X
n
,
n
≥
1
}
with general normalizing constants under a moment condition that
E
R
(
X
)
<
∞
, where
R
(
·
)
is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Stat Probab Lett 92:45–52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijn conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944–1946, 1995) on the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrative examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game. |
|---|---|
| ISSN: | 0894-9840 1572-9230 |
| DOI: | 10.1007/s10959-019-00973-2 |