Let $x_n$ be a sequence of independent random variables such that $P(x_n = 0) = 1 – \frac{1}{n}$
1) Does $x_n$ converge to $0$ almost surely
2) Does $x_n$ converge to $0$ in probability
3) Does $x_n$ converge to $0$ in $L_p$
It is clear for me that answer for 2) is yes: $P(|x_n| > \epsilon) \leq P(x_n \neq 0)$. Since the sequence of the last elements converges to $0$ we can conclude that $x_n$ converges to $0$ in probability.
I need some hints how to approach the remaining two questions. Thanks!
Best Answer
The answers to 1) and 3) are not necessarily.
For question $1$, consider a sequence of independent random variables $(X_n)$ such that $P(X_n=0)=1-n^{-1}$ and $P(X_n=1)=n^{-1}$. Argue using Borel-cantelli that $X_n=1$ infinitely often with probability one and hence the sequence does not converge a.s. to $0$.
For question $3$, consider the sequence of random variables $(X_n)$ such that $P(X_n=0)=1-n^{-1}$ and $P(X_n=n^{1/p})=1/n$ and show that $E|X_n|^p=1$ for all $n$.