a***n
发帖数: 40 | 1 (1) lim sup P(|Sn+j - Sn|>c)=O, lim is for n-->infinity, sup is over j>=1.
Is (1) the Cauchy criterion for Sn to converge in probablity?
(2) lim P{sup(|Sn+j - Sn|)>c}=O, lim is for n-->infinity, sup is over j>=1.
Is (2) the Cauchy criterion for Sn to converge almost surely? In other words,
can we take the limit inside and outside the probablity freely?
Thank you! | H****h
发帖数: 1037 | 2
Yes
,
Yes
No
【在 a***n 的大作中提到】
: (1) lim sup P(|Sn+j - Sn|>c)=O, lim is for n-->infinity, sup is over j>=1.
: Is (1) the Cauchy criterion for Sn to converge in probablity?
: (2) lim P{sup(|Sn+j - Sn|)>c}=O, lim is for n-->infinity, sup is over j>=1.
: Is (2) the Cauchy criterion for Sn to converge almost surely? In other words,
: can we take the limit inside and outside the probablity freely?
: Thank you!
| a***n
发帖数: 40 | 3 Thank you so much. If in addition, the probability is monotonic on the index n
, which is what the limit sends to inifinity, for example, An is nonincreasing
, then do we have P(lim An)=lim P(An)?
Thanks.
【在 H****h 的大作中提到】
:
: Yes
: ,
: Yes
: No
| H****h
发帖数: 1037 | | a***n
发帖数: 40 | 5 Thank you so much. If in addition, the probability is monotonic on the index n
, which is what the limit sends to inifinity, for example, An is nonincreasing
, then do we have P(lim An)=lim P(An)?
Thanks.
【在 H****h 的大作中提到】
: 他的第三个问题没有限制，所以回答否定。
| b********e
发帖数: 28 | 6 o, I agree.
【在 H****h 的大作中提到】
: 他的第三个问题没有限制，所以回答否定。
| H****h
发帖数: 1037 | 7
Yes
,
Yes
No
【在 a***n 的大作中提到】
: (1) lim sup P(|Sn+j - Sn|>c)=O, lim is for n-->infinity, sup is over j>=1.
: Is (1) the Cauchy criterion for Sn to converge in probablity?
: (2) lim P{sup(|Sn+j - Sn|)>c}=O, lim is for n-->infinity, sup is over j>=1.
: Is (2) the Cauchy criterion for Sn to converge almost surely? In other words,
: can we take the limit inside and outside the probablity freely?
: Thank you!
| b********e
发帖数: 28 | 8 Why can not take in? I think:
lim P(sup|Sn+j-Sn|>c)=lim E I(sup|Sn+j-Sn|>c)=E lim I (sup|Sn+j-Sn|>c)=
P(lim (sup|Sn+j-Sn|>c) by bounded convergent theorem.
,
Yes
No |
|
