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Miklós Schweitzer
1984 Miklós Schweitzer
10
10
Part of
1984 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1984- Problem 10
Source:
9/5/2016
10. Let
X
1
,
X
2
,
…
X_1, X_2, \dots
X
1
,
X
2
,
…
be independent random variables with the same distribution
P
(
X
i
=
1
)
=
P
(
X
i
=
−
1
)
=
1
2
(
i
=
1
,
2
,
…
)
P(X_i = 1) = P(X_i = -1)=\frac{1}{2}\qquad (i= 1, 2, \dots )
P
(
X
i
=
1
)
=
P
(
X
i
=
−
1
)
=
2
1
(
i
=
1
,
2
,
…
)
Define
S
0
=
0
,
S
n
=
X
1
+
X
2
+
⋯
+
X
n
(
n
=
1
,
2
,
…
S_0=0, Sn=X_1 +X_2+\dots +X_n \qquad (n=1, 2, \dots
S
0
=
0
,
S
n
=
X
1
+
X
2
+
⋯
+
X
n
(
n
=
1
,
2
,
…
),
ξ
(
x
,
n
)
=
∣
{
k
:
0
≤
k
≤
n
,
S
k
=
x
}
∣
(
x
=
0
,
±
1
,
±
2
,
…
\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right |\qquad (x=0, \pm 1, \pm 2, \dots
ξ
(
x
,
n
)
=
∣
{
k
:
0
≤
k
≤
n
,
S
k
=
x
}
∣
(
x
=
0
,
±
1
,
±
2
,
…
),and
α
(
n
)
=
∣
{
x
:
ξ
(
x
,
n
)
=
a
}
∣
(
n
=
0
,
1
,
…
\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right |\qquad (n=0,1,\dots
α
(
n
)
=
∣
{
x
:
ξ
(
x
,
n
)
=
a
}
∣
(
n
=
0
,
1
,
…
).Prove that
P
(
lim
inf
α
(
n
)
=
0
)
=
1
P(\lim \inf \alpha(n)=0) =1
P
(
lim
in
f
α
(
n
)
=
0
)
=
1
and that there is a number
0
<
c
<
∞
0<c<\infty
0
<
c
<
∞
such that
P
(
lim
inf
α
(
n
)
/
log
n
=
c
)
=
1
P(\lim \inf \alpha(n)/\log n=c) =1
P
(
lim
in
f
α
(
n
)
/
lo
g
n
=
c
)
=
1
(P.24) [P. Révész]
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