I have problems with the following task:
Let $X_{n} $ for $n=1,2,3...$ be independent random variables with distribution $B(n,\frac{1}{n})$ and define $Y_n=X_1...X_n$. Is $Y_n$ convergent in $L^1$?
My attempts:
I know that $Y_n$ is martingale. I tried to show that $Y_n$ are uniformly integrable but I'm not even sure if it is true. Thank you for some help.