Assume that $\{\tau_n: n \in \mathbb{N}\} $ is a sequence of stopping times with respect to some filtration such that $P[\tau_n < \infty] = 1. $ Is that true that there must exist a sequence of constants $\{m_n: n \in \mathbb{N}\} \in \mathbb{R}_+ $ such that $$ P[\tau_n > m_n] \to 0 $$ as $n \to \infty$?
I realize that this is clearly true for the case in which one deals with a single finite stopping time, $\tau, $ but for a general sequence? Thank you in advance for any help.
Maurice
For each non-negative random variable $X$ such that $\mathbb P\{X\lt+\infty\}=1$, we have $\lim_{R\to +\infty}\mathbb P\{X\gt R\}=0$. With this argument, for each $n$, you can choose $m_n$ such that $\mathbb P\{\tau_n\gt m_n\}\lt 1/n$.