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

Hi guys,

Any chance anyone could help me think of a function as in probability but it does not convergence almost surely nor in ?

Thanks in advance!

YChana

Hello,

If I'm not mistaking, you can have a look at sequence of independent rv's where

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)

how about P(X_n=n)=1/n and P(X_n=0)=1-1/n

Quote:

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Originally Posted by matheagle

how about P(X_n=n)=1/n and P(X_n=0)=1-1/n

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It's the same for any sequence following a Bernoulli distribution actually :)