Let $(X_n)$ and $(Y_n)$ be tail equivalent random variables i.e. $\sum_{i=1}^{\infty}\mathbb P(X_i\neq Y_i)<\infty$

Show that $\sum_{n=1}^{\infty}X_n$ and $\sum_{n=1}^{\infty}Y_n$ converge or diverge together.

I guess this is not an hard task. I know that $\mathbb P(X_i\neq Y_i)$ is a zero sequence. But the my problem is how to use this here. Also: which norm are we using here?


1 Answer 1


By the 1st Borel Cantelli lemma, $P(X_i \neq Y_i \,\,i.o.)=0$. So for almost every $\omega$, $\exists \,N(\omega)$ such that for $n \geq N(\omega)$, $X_n(\omega)=Y_n(\omega)$. Since the convergence and divergence of any series of real numbers depends only on the tail, for almost every $\omega$, $\sum_{i=1}^\infty X_i(\omega)$ and $\sum_{i=1}^\infty Y_i(\omega)$ either both diverge or both converge.


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