1
$\begingroup$

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 $n\geq n_0$,

$$P(|X_n-Y_n|>\epsilon)<\delta$$

or, in other words,

$$\lim_{n\rightarrow\infty}P(|X_n-Y_n|>\epsilon)=0$$

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

$$\lim_{n\rightarrow\infty}P(X_n>s_n)=1$$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

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.