## Result as to uniformly integrable sequence

ダウンロード数10

• ページ数 : 3ページ
• 会員550円 | 非会員660円

### 資料紹介

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...

## ats0307の資料(31)

### コメント2件

maxwell123
good!
2007/01/15 0:42 (17年8ヶ月前)

831hiro
good
2007/01/19 23:30 (17年8ヶ月前)

### コメント追加

コメントを書込むには会員登録するか、すでに会員の方はログインしてください。