Let $\{a_n\}$ be a number sequence that converges to a limit $a\in\mathbb{R}$, and fix $\varepsilon>0$. Then there exists some integer $m\in\mathbb{N}$ such that for all $n\geq m$ it holds that $|a_n-a_m|<\varepsilon$.
Now let $(\Omega,\mathbb{P})$ be a probability space, and let $\{X_n\}$ be a bounded, non-negative martingale. Thus, it converges a.s to a finite random variable $X_\infty$. By definition, this means that with probability 1, $\lim_{n\to\infty} X_n=X_\infty$.
I'm trying to translate the Cauchy-like logic from number sequences to converging martingales. Intuitively, one (me) might say immediately that, fixing $\varepsilon>0$ there exists some integer $m\in\mathbb{N}$ such that for all $n\geq m$, $|X_n-X_m|< \varepsilon$.
- Is that $m$ then a random variable itself?
- If so, it is not necessarily a stopping time. What can one say on $X_m$ then? Can I define it as a random variable as well?
Thanks in advance for all types of comments and inputs!