0
$\begingroup$

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!

$\endgroup$
1
$\begingroup$

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 $n$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.