An almost sure central limit theorem for products of sums under association
Let { X n , n ⩾ 1 } be a strictly stationary positively or negatively associated sequence of positive random variables with E X 1 = μ > 0 and Var ( X 1 ) = σ 2 < ∞ . Denote S n = ∑ i = 1 n X i and γ = σ / μ the coefficient of variation. Under suitable conditions, we show that ∀ x lim n → ∞ 1 l...
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Published in | Statistics & probability letters Vol. 78; no. 4; pp. 367 - 375 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier B.V
01.03.2008
Elsevier |
Series | Statistics & Probability Letters |
Subjects | |
Online Access | Get full text |
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Summary: | Let
{
X
n
,
n
⩾
1
}
be a strictly stationary positively or negatively associated sequence of positive random variables with
E
X
1
=
μ
>
0
and
Var
(
X
1
)
=
σ
2
<
∞
. Denote
S
n
=
∑
i
=
1
n
X
i
and
γ
=
σ
/
μ
the coefficient of variation. Under suitable conditions, we show that
∀
x
lim
n
→
∞
1
log
n
∑
k
=
1
n
1
k
I
∏
j
=
1
k
S
j
k
!
μ
k
1
/
(
γ
σ
1
k
)
⩽
x
=
F
(
x
)
a
.
s
.,
where
σ
1
2
=
1
+
2
σ
2
∑
j
=
2
∞
Cov
(
X
1
,
X
j
)
,
F
(
·
)
is the distribution function of the random variable
e
2
N
, and
N
is a standard normal random variable. This extends the earlier work on independent, positive random variables (see Khurelbaatar and Rempala [2006. A note on the almost sure limit theorem for the product of partial sums. Appl. Math. Lett. 19, 191–196]). |
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ISSN: | 0167-7152 1879-2103 |
DOI: | 10.1016/j.spl.2007.07.009 |