Problem: Let <mo fence="false" stretchy="false">{ X i </msub> <msubsup>

Ayanna Trujillo 2022-06-24 Answered
Problem:
Let { X i } i = 1 be a sequence of random variables on a probability space ( Ω , F , P ) such that lim i X i = X  a.e. a.e. Show that if sup i E ( X i 2 ) < , then E ( X 2 ) < .
My Attempt:
I will try to explain as best I can. First, I have a version of Fatou's Lemma stating that if { X i } i = 1 is a sequence of non-negative random variables, then E ( lim inf i X i ) lim inf i E ( X i ).
If we let Y i = X i 2 then I have a sequence of non-negative random variables to work with. One concern of mine is this: can I assume that lim i X i 2 = X 2 ? I feel like that's necessary for for what I've written below to work.
If I can make that assumption then we have
E ( X 2 ) = E ( lim inf i X i 2 )  (Is this justified?) lim inf i E ( X i 2 )  (application of Fatou's Lemma) sup i E ( X i 2 )  (property of real numbers) <  (by assumption) .
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Answers (1)

hildiadau0o
Answered 2022-06-25 Author has 21 answers
Yes, your first step is justified. By the continuous mapping theorem, we have that if X n X almost surely, then g ( X n ) g ( X ) almost surely for any continuous function. If we set g ( x ) = x 2 , then
X 2 = lim n X n 2 = lim inf n X n 2 a.s
where the last equality follows simply because the limit exists. The rest follows by taking expectations and applying Fatou's lemma.

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