Comparison of the elements of two sequences and their distribution limits

Question

Assume that we have two sequences of positive integrable r.v. $X_{n}$ and $Y_{n}$ defined on the same probability space. Next assume that $$ X_{n}\stackrel{d}{\to}X $$ and $$ Y_{n}\stackrel{d}{\to}Y $$ and let $E[X] < E[Y]$. Does this mean then $$ P[\liminf_{n\to\infty}\{X_{n} \leq Y_{n}\}] =1 $$ or $$ P[X_{n}\leq Y_{n}] \to 1 \text{ as } n\to\infty $$ ?

Answer 1

No. You can construct a counterexample fairly easily; consider a case where $X_n = X$ and $Y_n = Y$ for all $n$; then, your task is only to construct variables $X, Y$ where $\mathbb E[X] < \mathbb E[Y]$, and yet $X > Y$ with positive probability. If you want, you could even do this in such a way that $X$ and $Y$ are independent.