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

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• April 13th 2011, 04:30 AM
ychana
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
• April 13th 2011, 06:58 AM
Moo
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)
• April 15th 2011, 11:19 PM
matheagle
how about P(X_n=n)=1/n and P(X_n=0)=1-1/n
• April 16th 2011, 05:17 AM
Moo
Quote:

Originally Posted by matheagle
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 :)