Proof of Helly-Bray Theorem, Continuity Theorem and Cramer Wold Theorem

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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 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

    資料の原本内容( この資料を購入すると、テキストデータがみえます。 )

    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..

    コメント6件

    kagochii38 購入
    興味深かった。
    2006/12/07 16:29 (10年前)

    nicole0218 購入
    参考にさせていただきます
    2006/12/08 17:22 (10年前)

    blackbox 購入
    a
    2006/12/08 22:41 (10年前)

    syu_30 購入
    参考になりました
    2006/12/25 0:44 (10年前)

    kanotch 購入
    参考になりました
    2006/12/29 11:39 (10年前)

    shu600507 購入
    good!
    2007/01/02 12:38 (9年11ヶ月前)

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