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 ) &lt; ∞, then E ( X 2 ) &lt; ∞.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) &lt; ∞ (by assumption) .

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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    )  &amp;lt;  ∞, then   E  (      X    2    )  &amp;lt;  ∞.My Attempt:I will try to explain as best I can. First, I have a version of Fatou&#039;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&#039;s necessary for for what I&#039;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&#039;s Lemma)                                          ≤                  sup          i                E        (                  X          i          2                )         (property of real numbers)                                          &amp;lt;        ∞         (by assumption)        .

hildiadau0o · Accepted Answer

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.swhere the last equality follows simply because the limit exists. The rest follows by taking expectations and applying Fatou&#039;s lemma.

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

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