Let $a_1,a_2,\dots $ be a sequence of random variables. They are not assumed i.i.d. Say that $$\sum\limits_{k=1}^{\infty}\mathbb{E}[|a_k|^2] := A < \infty.$$
For $\epsilon> 0$, define the stopping time$$T_{\epsilon} = \inf\{ k \mid |a_k| < \epsilon\}.$$ Does $T_{\epsilon}$ necessarily have finite expectation? and can I bound $\mathbb{E}[T_{\epsilon}]$ in terms of $A$ and $\epsilon$?
In the deterministic case, this is straightforward:
If $T_{\epsilon} \geq \lceil \frac{A}{\epsilon^2}\rceil$ then $$\sum\limits_{k=1}^{\infty}|a_i|^{2} > \frac{A}{\epsilon^2} \epsilon^2 = A,$$ a contradiction. Can any similar result be proven for the stochastic case?
$E\sum _{k=1}^{T_\epsilon -1} a_k ^{2} \leq E\sum _{k=1}^{\infty} a_k ^{2} = A$. This gives $E\sum _{k=1}^{T_\epsilon -1} \epsilon^{2} \leq A$. Hence $ET_{\epsilon} \leq 1+\frac A {{\epsilon}^{2}}$. ( $T_\epsilon -1$ need not be an integer, so an obvious modification is needed, but I will that to you).