Re: Help: Exchange of limit and expectation


Herman Rubin wrote:
In article <1152972077.474858.29030@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Yecloud <yecloud@xxxxxxxxx> wrote:
Hi, all,

Please give me a help. If I have a sequence of random variables {X_n},
X_n>=0 and
if I know the limit of X_n and the limit of E(X_n) are bounded (where
E(X_n) is monotonically
increasing), what's the condition for lim(E(X_n))=E(lim(X_n))? or if it
always holds, how to
prove it?

If the X_n are monotonically increasing, this is merely
the monotone convergence theorem. If not, one needs a
uniform integrability condition, which is needed to
prevent P(E_n) -> 0 and E(X_n*I_E_n) not going to 0.

Thanks! Actually, in my problem, X_n is a function of two random
variables
(s,t), i.e., X_n = f_n(s)*exp(-t). for each s, f_n(s) is montonically
increasing with n. Do you think the monotone convergence theorem could
be used here?

Thanks a lot.
-Cloud






--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558

.

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