Hi guys,

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

Thanks in advance!

YChana

- Apr 13th 2011, 04:30 AMychanaConvergence in probability but no convergence in L^p nor A.S.
Hi guys,

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

Thanks in advance!

YChana - Apr 13th 2011, 06:58 AMMoo
Hello,

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

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) - Apr 15th 2011, 11:19 PMmatheagle
how about P(X_n=n)=1/n and P(X_n=0)=1-1/n

- Apr 16th 2011, 05:17 AMMoo