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An extension of a theorem of Mikosch
详细信息
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作者:
Yuye Zou
a
;
Xiangdong
Li
u
b
;
t
li
uxd@jnu.edu.cn" class="auth_mail" title="E-mail the corresponding author
关键词:
60F15
;
60G50
刊名:Statistics & Probabi
li
ty Letters
出版年:2017
出版时间:January 2017
年:2017
卷:120
期:Complete
页码:81-86
全文大小:383 K
文摘
Let
lick to view the MathML source">0<α≤2
0
<
α
≤
2
. Let
lick to view the MathML source">N
d
N
d
be the
lick to view the MathML source">d
d
-dimensional lattice equipped with the coordinate-wise partial order
lick to view the MathML source">≤
≤
, where
lick to view the MathML source">d≥1
d
≥
1
is a fixed integer. For
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li
mgeid="1-s2.0-S0167715216301791-si6.gif">
lign:bottom" width="150" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167715216301791-si6.gif">
n
=
(
n
1
,
&hel
li
p;
,
n
d
)
∈
N
d
, define
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li
mgeid="1-s2.0-S0167715216301791-si7.gif">
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|
n
|
=
∏
i
=
1
d
n
i
. Let
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li
mgeid="1-s2.0-S0167715216301791-si8.gif">
lign:bottom" width="105" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0167715216301791-si8.gif">
{
X
,
X
n
;
n
∈
N
d
}
be a field of independent and identically distributed real-valued random variables. Set
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li
mgeid="1-s2.0-S0167715216301791-si9.gif">
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S
n
=
∑
k
≤
n
X
k
,
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li
mgeid="1-s2.0-S0167715216301791-si10.gif">
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n
∈
N
d
and write
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li
mgeid="1-s2.0-S0167715216301791-si11.gif">
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log
x
=
log
e
(
e
∨
x
)
,
x
≥
0
. This note is devoted to an extension of a strong
li
mit theorem of Mikosch (1984). By applying an idea of Li and Chen (2014) and the classical Marcinkiewicz–Zygmund strong law of large numbers for random fields, we obtain necessary and sufficient conditions for
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li
mgeid="1-s2.0-S0167715216301791-si12.gif">
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li
m sup
n
|
S
n
|
(
log
|
n
|
)
−
1
=
Δ
li
m
m
→
∞
sup
|
n
|
≥
m
|
S
n
|
(
log
|
n
|
)
−
1
=
e
1
/
α
almost surely .
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