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Math Help - Some problems

  1. #1
    Member
    Joined
    Jan 2006
    From
    Gdansk, Poland
    Posts
    117

    Some problems

    1

    Prove that if X_n \rightarrow X in probability than X_n \Rightarrow X (in distribution).

    My approach:

    F_{X_n}(t) = P\{X_n \le t \} =
    = P\{X_n \le t,\quad X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \}
    \le  P\{X \le t+\epsilon \} + P\{X_n \le t, \quad X > t+\epsilon \}

    Because X_n \rightarrow X in probability \lim_{n \rightarrow \infty} P\{X_n \le t, \quad X > t+\epsilon \} = 0

    Thus:
    \limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t+\epsilon)

    Hence: \limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t)

    I was unable to prove the other inequality unless X is absolutly continuous.
    Does anybody know how to do that?


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    2.

    Let Y_n be random variables with characteristic functions \phi_n, n = 1, 2,...
    Prove that Y_n \Rightarrow 0 (in distribution) if there exists \delta > 0 such that \lim_{n\rightarrow\infty} \phi_n(t) = 1 for |t| < \delta.

    I was unable to solve it. The only reasonable idea I had was to assume that all moments of the variables exist and prove thay all converge to 0 (than convergense of distribution follows, I think)

    Can you help me with this one too?
    Last edited by albi; June 9th 2008 at 10:47 AM. Reason: Error in proof
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  2. #2
    Member
    Joined
    Jan 2006
    From
    Gdansk, Poland
    Posts
    117

    1st problem

    Finally I solved the first one. I thought X_n \Rightarrow X (in distribution) means that F_{X_n}(t) \rightarrow F_X(t) FOR ALL t.

    While it is the real definition is F_{X_n}(t) \rightarrow F_X(t) FOR ALL t WHERE F_X is CONTINUOUS.

    So the solution is:

    F_{X}(t - \epsilon) = P\{X \le t - \epsilon \} =
    = P\{X \le t-\epsilon,\quad X_n \le t \} + P\{X \le t-\epsilon, \quad X_n > t \}
    \le P\{X_n \le t \} + P\{X \le t-\epsilon, \quad X_n > t \}

    Because X_n \rightarrow X in probability \lim_{n \rightarrow \infty} P\{X \le t-\epsilon, \quad X_n > t \} = 0

    Thus:
    \liminf_{n \rightarrow \infty}F_{X_n}(t) \ge F_{X}(t-\epsilon)

    Hence (FX is continuous in t): \liminf_{n \rightarrow \infty}F_{X_n}(t) \ge F_{X}(t)

    Finally

     \limsup_{n \rightarrow \infty}F_{X_n}(t) \le F_{X}(t) \le \liminf_{n \rightarrow \infty}F_{X_n}(t)

    Which solves the problem!

    The second one is still unsolved!
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