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Miklós Schweitzer
1970 Miklós Schweitzer
11
Miklos Schweitzer 1970_11
Miklos Schweitzer 1970_11
Source:
October 22, 2008
limit
probability and stats
Problem Statement
Let
ξ
1
,
ξ
2
,
.
.
.
\xi_1,\xi_2,...
ξ
1
,
ξ
2
,
...
be independent random variables such that
E
ξ
n
=
m
>
0
E\xi_n=m>0
E
ξ
n
=
m
>
0
and
Var
(
ξ
n
)
=
σ
2
<
∞
(
n
=
1
,
2
,
.
.
.
)
.
\textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .
Var
(
ξ
n
)
=
σ
2
<
∞
(
n
=
1
,
2
,
...
)
.
Let
{
a
n
}
\{a_n \}
{
a
n
}
be a sequence of positive numbers such that
a
n
→
0
a_n\rightarrow 0
a
n
→
0
and
∑
n
=
1
∞
a
n
=
∞
\sum_{n=1}^{\infty} a_n= \infty
∑
n
=
1
∞
a
n
=
∞
. Prove that
P
(
lim
n
→
∞
∑
k
=
1
n
a
k
ξ
k
=
∞
)
=
1.
P \left( \lim_{n\rightarrow \infty} %Error. "diaplaymath" is a bad command. \sum_{k=1}^n a_k \xi_k =\infty \right)=1.
P
(
n
→
∞
lim
k
=
1
∑
n
a
k
ξ
k
=
∞
)
=
1.
P. Revesz
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