a***n
发帖数: 40 | 1 If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
and Xn is uniformly bounded,i.e., |Xn| < k,
then the infinite sum of Xn converges almost surely?
Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
probably from a null set, Xn(w) is always 0 after a certain number of terms?
Thanks a lot. | H****h
发帖数: 1037 | 2
No
No
【在 a***n 的大作中提到】
: If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
: and Xn is uniformly bounded,i.e., |Xn| < k,
: then the infinite sum of Xn converges almost surely?
: Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
: probably from a null set, Xn(w) is always 0 after a certain number of terms?
: Thanks a lot.
| a***n
发帖数: 40 | 3 Thanks a lot. Can you please explain why?
【在 H****h 的大作中提到】
:
: No
: No
| a***n
发帖数: 40 | 4 I just found that I made a mistake. the number c is arbitrary for these two
questions:
If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
and Xn is uniformly bounded,i.e., |Xn| < k,
then the infinite sum of Xn converges almost surely?
Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
probably from a null set, Xn(w) is always 0 after a certain number of terms?
If we let c goes to 0, are they true? Thanks.
【在 a***n 的大作中提到】
: If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
: and Xn is uniformly bounded,i.e., |Xn| < k,
: then the infinite sum of Xn converges almost surely?
: Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
: probably from a null set, Xn(w) is always 0 after a certain number of terms?
: Thanks a lot.
| H****h
发帖数: 1037 | 5 你还是自己考虑一下吧。
【在 a***n 的大作中提到】
: I just found that I made a mistake. the number c is arbitrary for these two
: questions:
: If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
: and Xn is uniformly bounded,i.e., |Xn| < k,
: then the infinite sum of Xn converges almost surely?
: Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
: probably from a null set, Xn(w) is always 0 after a certain number of terms?
: If we let c goes to 0, are they true? Thanks.
| D*******a
发帖数: 3688 | 6
The infinite sum cannot converge a.s.
Suppose your prob space is [0,1].
Xn=1 if 0
For all M, as long as 0M
【在 a***n 的大作中提到】
: I just found that I made a mistake. the number c is arbitrary for these two
: questions:
: If Xn converges to 0 almost surely, i.e., P(|Xn|>c, i.o.) = 0, for any c>0;
: and Xn is uniformly bounded,i.e., |Xn| < k,
: then the infinite sum of Xn converges almost surely?
: Does P(|Xn|>c, i.o.) = 0, for any c>0, mean that for any event w, except
: probably from a null set, Xn(w) is always 0 after a certain number of terms?
: If we let c goes to 0, are they true? Thanks.
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