資料紹介

Proof of Helly-Bray Theorem, Continuity Theorem
and Cram´er -Wold Theorem
Theorem(Helly-Bray)Let g(x) be bounded,continuous function and assume thatfFn(x)g is a sequence of
uniformly bounded,non decreasing distribution functions which converges to F(x) at all points of open
interval( ; ),then
R
g(x)dFn(x)converges to
R
g(x)dF(x).In other words,if Xn converges in distribution to
X,then
R
g(x)dFn(x)converges to
R
g(x)dF(x).
Proof
Let us consider the Stiljes integral ofg(x) and choose two continuity pointsa,b (a < b) ofF(x) and write the
integral as
Z
g(x)dFn(x)
Z
g(x)dF(x) =
Z a
g(x)(dFn dF)(x)+
Z b
a
g(x)(dFn dF)(x)+
Z
b
g(x)(dFn dF)(x)
=
hZ a
g(x)dFn(x)
Z a
g(x)dF(x)
i
+
hZ b
a
g(x)dFn(x)
Z b
a
g(x)dF(x)
i
+
hZ
b
g(x)dFn(x)
Z
b
g(x)dF(x)
i
(1)
Letjg(x)j < c< . Then the absolute value of the first integral in the second line in Eq.(1) satisfies
Z a
g(x)dFn(x)
Z a
g(x)dF(x)
< c
Z a
dFn(x)+ c
Z a
dF(x) = c[Fn(a)+ F(a)] (2)
The last equality holds from the assumptions that sinceF is a distribution function thenFn( ) = F( ) = 0.
Ifa is sufficiently small,F(a) is small and so isFn(a) for alln > N. Hence for suitablea and N, we can let
c[Fn(a)+ F(a)]< e. Similarly as to the third term of the second line in Eq.(1), we can write it as, noting that
Fn(+ ) = F(+ ) = 1,
Z
b
g(x)dFn(x)
Z
b
g(x)dF(x)
< c
Z
b
dFn(x)+ c
Z
b
dF(x) = 2c c[Fn(b)+ F(b)] (3)
The same logic gives that2c c[Fn(b)+ F(b)]< e for sufficienly largeb and N. In the finite interval[a;b],g(x)
is uniformaly continuous. L

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Proof of Helly-Bray Theorem, Continuity Theorem
and Cram´er -Wold Theorem
Theorem(Helly-Bray)Let g(x) be bounded,continuous function and assume thatfFn(x)g is a sequence of
uniformly bounded,non decreasing distribution functions which converges to F(x) at all points of open
interval( ; ),then
R
g(x)dFn(x)converges to
R
g(x)dF(x).In other words,if Xn converges in distribution to
X,then
R
g(x)dFn(x)converges to
R
g(x)dF(x).
Proof
Let us consider the Stiljes integral ofg(x) and choose two continuit...

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