If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random variables and $X,Y$ are two random variables, then what is $\lim\limits_{n \to \infty} P(X_n\leq Y_n)$?

We can consider $\lim\limits_{n \to \infty}P(X_n\leq Y_n)=\lim\limits_{n \to \infty}P(\frac{X_n}{Y_n}\leq 1 )$. Can we say anything about convergence of $X_n/Y_n$? Can you please help?

  • 3
    $\begingroup$ Without knowledge of the joint distribution, no chance. $\endgroup$ – zhoraster May 15 '16 at 20:34

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