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Miklós Schweitzer
1979 Miklós Schweitzer
11
Miklos Schweitzer 1979_11
Miklos Schweitzer 1979_11
Source: double sequence of random variables
January 28, 2009
probability
probability and stats
Problem Statement
Let
{
ξ
k
ℓ
}
k
,
ℓ
=
1
∞
\{\xi_{k \ell} \}_{k,\ell=1}^{\infty}
{
ξ
k
ℓ
}
k
,
ℓ
=
1
∞
be a double sequence of random variables such that
E
(
ξ
i
j
ξ
k
ℓ
)
=
O
(
log
(
2
∣
i
−
k
∣
+
2
)
log
(
2
∣
j
−
ℓ
∣
+
2
)
2
)
(
i
,
j
,
k
,
ℓ
=
1
,
2
,
…
)
.
\Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .
E
(
ξ
ij
ξ
k
ℓ
)
=
O
(
lo
g
(
2∣
j
−
ℓ
∣
+
2
)
2
lo
g
(
2∣
i
−
k
∣
+
2
)
)
(
i
,
j
,
k
,
ℓ
=
1
,
2
,
…
)
.
Prove that with probability one,
1
m
n
∑
k
=
1
m
∑
ℓ
=
1
n
ξ
k
ℓ
→
0
as
max
(
m
,
n
)
→
∞
.
\frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .
mn
1
k
=
1
∑
m
ℓ
=
1
∑
n
ξ
k
ℓ
→
0
as
max
(
m
,
n
)
→
∞
.
F. Moricz
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