Result as to uniformly integrable sequence

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    Result as to Uniformly Integrable Sequence
    Result Suppose that there exists r >1 and M < 1 such that EjXtXt kj
    r < M for all t , k and
    that
    P 1
    j = 1 jhj j < 1 . Define Yt =
    P 1
    j = 1 hj Xt j . Then fYtYt kg is uniformaly integrable for
    any integer k. This implies that a product of uniformly integrable sequence is also uniformly
    integrable.
    Proof
    Firstly the productYtYt k takes the form of
    YtYt k =
    1X
    i= 1
    hiXt i
    1X
    j = 1
    hj Xt k j =
    1X
    i= 1
    1X
    j = 1
    hihj Xt iXt k j
    According to the definition of uniformly integrability, we consider whether or notE[jYtYt kjfjYtYt k j cg]
    converges to zero, wherefjYtYt k j cg equals to one if the argument is met and equals to zero otherwise.
    Now we have
    E
    h
    jYtYt kj fjYtYt k j cg
    i
    = E
    h
    1X
    i= 1
    hiXt i
    1X
    j = 1
    hj Xt k j
    fjYtYt k j cg
    i
    = E
    h
    1X
    i= 1
    1X
    j = 1
    hihj Xt iXt k j
    fjYtYt k j cg
    i
    E
    h 1X
    i= 1
    1X
    j = 1
    jhihj jjXt iXt k j j fjYtYt k j cg
    i
    (1)
    Here we can change the operation of expectaion and summation. In order to conduct this changing,
    we need to confirm thatE[jXt iXt k j j fjYtYt k j cg] is bounded and fhihj g is absolutely summable,
    however, both of which are proved lastly(). Then we can rewrite Eq.(1) as
    E
    h
    jYtYt kj fjYtYt k j cg
    i
    E
    h 1X
    i= 1
    1X
    j = 1
    jhihj jjXt iXt k j j fjYtYt k j cg
    i
    =
    1X
    i= 1
    1X
    j = 1
    jhihj jE
    jXt iXt k j j fjYtYt k j cg
    (2)
    As to the expectation term, using H¨older’s inequality, we can get
    E
    jXt iXt k j j fjYtYt k j cg
    EjXt iXt k j j
    r
    1=r
    E[fjYtYt k j cg]
    r=(r 1)
    (r 1)=r
    =
    EjXt iXt k j j
    r
    1=r
    E[fjYtYt k j

    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。 )

    Result as to Uniformly Integrable Sequence
    Result Suppose that there exists r >1 and M < 1 such that EjXtXt kj
    r < M for all t , k and
    that
    P 1
    j = 1 jhj j < 1 . Define Yt =
    P 1
    j = 1 hj Xt j . Then fYtYt kg is uniformaly integrable for
    any integer k. This implies that a product of uniformly integrable sequence is also uniformly
    integrable.
    Proof
    Firstly the productYtYt k takes the form of
    YtYt k =
    1X
    i= 1
    hiXt i
    1X
    j = 1
    hj Xt k j =
    1X
    i= 1
    1X
    j = 1
    hihj Xt iXt k j
    According to the definition of..

    コメント2件

    maxwell123 購入
    good!
    2007/01/15 0:42 (9年11ヶ月前)

    831hiro 購入
    good
    2007/01/19 23:30 (9年11ヶ月前)

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