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
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
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)