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Math Help - Convergence in probability but no convergence in L^p nor A.S.

  1. #1
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    Convergence in probability but no convergence in L^p nor A.S.

    Hi guys,

    Any chance anyone could help me think of a function X_n \rightarrow X as n \rightarrow \infty in probability but it does not convergence almost surely nor in L^p?

    Thanks in advance!

    YChana
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  2. #2
    Moo
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    Hello,

    If I'm not mistaking, you can have a look at (X_n)_n sequence of independent rv's where P(X_n=0)=\frac 1n=1-P(X_n=1)

    It does converge in probability, but doesn't converge in Lp. As for the almost sure converge, we'll have to use Borel-Cantelli (look here for proving that part)
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  3. #3
    MHF Contributor matheagle's Avatar
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    how about P(X_n=n)=1/n and P(X_n=0)=1-1/n
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  4. #4
    Moo
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    Quote Originally Posted by matheagle View Post
    how about P(X_n=n)=1/n and P(X_n=0)=1-1/n
    It's the same for any sequence following a Bernoulli distribution actually
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