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Miklós Schweitzer
1996 Miklós Schweitzer
10
10
Part of
1996 Miklós Schweitzer
Problems
(1)
exchangeable random variables and order statistics
Source: miklos schweitzer 1996 q10
10/11/2021
Let
Y
1
,
.
.
.
,
Y
n
Y_1 , ..., Y_n
Y
1
,
...
,
Y
n
be exchangeable random variables, ie for all permutations
π
\pi
π
, the distribution of
(
Y
π
(
1
)
,
…
,
Y
π
(
n
)
)
(Y_{\pi (1)}, \dots, Y_{\pi (n)} )
(
Y
π
(
1
)
,
…
,
Y
π
(
n
)
)
is equal to the distribution of
(
Y
1
,
.
.
.
,
Y
n
)
(Y_1 , ..., Y_n)
(
Y
1
,
...
,
Y
n
)
. Let
S
0
=
0
S_0 = 0
S
0
=
0
and
S
j
=
∑
i
=
1
j
Y
i
j
=
1
,
…
,
n
S_j = \sum_{i = 1}^j Y_i \qquad j = 1,\dots,n
S
j
=
i
=
1
∑
j
Y
i
j
=
1
,
…
,
n
Denote
S
(
0
)
,
.
.
.
,
S
(
n
)
S_{(0)} , ..., S_{(n)}
S
(
0
)
,
...
,
S
(
n
)
by the ordered statistics formed by the random variables
S
0
,
.
.
.
,
S
n
S_0 , ..., S_n
S
0
,
...
,
S
n
. Show that the distribution of
S
(
j
)
S_{(j)}
S
(
j
)
is equal to the distribution of
max
0
≤
i
≤
j
S
i
+
min
0
≤
i
≤
n
−
j
(
S
j
+
i
−
S
j
)
\max_{0 \le i \le j} S_i + \min_ {0 \le i \le n-j} (S_{j + i} -S_j)
max
0
≤
i
≤
j
S
i
+
min
0
≤
i
≤
n
−
j
(
S
j
+
i
−
S
j
)
.
probability and stats