Suppose I have two sequences of positive random variables $(X_n)$ and $(Y_n)$ and the following is true: for all $\epsilon>0$ and $\delta>0$, there exists $n_0$ such that for all $n\geq n_0$,


or, in other words,


While this resembles convergence in probability, this is only a resemblance as it's a sequence "converging" to another sequence (obviously, this does show convergence in probability to zero of a sequence $(Z_n)$ where $Z_n=X_n-Y_n$).

Now also suppose that


where $s_n$ is a non-decreasing positive sequence.

Does the above imply that $\lim_{n\rightarrow\infty}P(Y_n>s_n)=1$? That is, can one interchange $X_n$ and $Y_n$ in the limit above? If not, what is the necessary condition for the exchange of sequences to be valid?

I suspect that the implication does not hold, but I am confused with the limits... any help/hints would be appreciated.


No. You could define $X_n$ to equal $s_n + \frac1n$ almost surely and $Y_n$ to equal $s_n - \frac1n$ almost surely, for example.


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