a***n
发帖数: 40 | 1 If Xn converges to X in probability, i.e.,
for any e>0, lim P( |Xn - X| < e ) = 1,
can we let e -> 0, and conclude that lim P( Xn = X ) = 1 ?
I really doubt it, but cannot see why not? Thanks for your help. | H****h
发帖数: 1037 | 2
No
set
No
so
【在 a***n 的大作中提到】
: If Xn converges to X in probability, i.e.,
: for any e>0, lim P( |Xn - X| < e ) = 1,
: can we let e -> 0, and conclude that lim P( Xn = X ) = 1 ?
: I really doubt it, but cannot see why not? Thanks for your help.
| c*******n
发帖数: 718 | 3 Xn=I(0,1)*(1-1/n)
X=I(0,1)
Then lim P(|Xn-X|
but obviously, Xn<>X everywhere i.e. P(Xn=X)=0 for any n
【在 a***n 的大作中提到】
: If Xn converges to X in probability, i.e.,
: for any e>0, lim P( |Xn - X| < e ) = 1,
: can we let e -> 0, and conclude that lim P( Xn = X ) = 1 ?
: I really doubt it, but cannot see why not? Thanks for your help.
| a***n
发帖数: 40 | 4 Thank you. Another question is that if Xn converges to X almost surely, then
lim P(Xn=X)=1. Is it correct? My understanding of convergence almost surely is
that after finite many terms, Xn and X and the same except probably on a set
of measure 0. Is that correct?
I am trying to understand the various modes of convergence, and my questions
may be pretty naive. But it clarifies some of my doubts, and your helps are so
much appreciated.
【在 c*******n 的大作中提到】
: Xn=I(0,1)*(1-1/n)
: X=I(0,1)
: Then lim P(|Xn-X|: but obviously, Xn<>X everywhere i.e. P(Xn=X)=0 for any n
| H****h
发帖数: 1037 | 5
No
set
No
so
【在 a***n 的大作中提到】
: Thank you. Another question is that if Xn converges to X almost surely, then
: lim P(Xn=X)=1. Is it correct? My understanding of convergence almost surely is
: that after finite many terms, Xn and X and the same except probably on a set
: of measure 0. Is that correct?
: I am trying to understand the various modes of convergence, and my questions
: may be pretty naive. But it clarifies some of my doubts, and your helps are so
: much appreciated.
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