I was studying SPRT (Sequential Probability Ratio Tests) and there was a section (in an online article I was reading) which proved optimality of SPRT using some approximations. Unfortunately, this made the proof weak and I wanted to know if I could come up with an improved proof. Without going into many details, the following is my question:
Let $\{Z_i\}$ be iid random variables ($\sim P_Z$) with finite moments and positive mean. Let $S_n = \sum_{i=1}^nZ_i$. Assume $S_0 = 0 ~ a.s$. Given $-\infty < a<0 <b<\infty$, define $N$ as $$N = \inf\{n\geq 1:S_n \notin (a,b)\}$$ I need to find useful bounds on $E[N]$ in terms of $P_Z$ (and some moments), $a$, $b$ and perhaps other parameters. From Wald's Lemma, we have $E[S_N] = E[N]E[Z_1]$, so bounds on $E[S_N]$ would work as well.
I have seen analysis of simple random walks, Gambler Ruins etc. but could not particularize the results to this case. If someone could provide good references to results or proofs, that would be appreciated.
Let me know if more information is required.
Update: I tried this problem for a while and I got the following. In order to compute $E[N]$, it helps to characterize $P(N\leq n)$. To this end, we have $$P(N\leq n) = P\left(\bigcup_{k=1}^n\{S_k \geq b\}\cup \{S_k \leq a\}\right) $$ $$= P\left(\bigcup_{k=1}^n\{S_k \geq b\}\right)+ P\left(\bigcup_{k=1}^n\{S_k \leq a\}\right)$$ $$\geq P\left(S_n \geq b\right)+ P\left(S_n \leq a\right)$$ From this, $P(N \geq n+1) \leq P\left(S_{n} \in (a,b)\right)$. This will imply $$E[N] \leq 1 + \sum_{n=1}^\infty P\left(S_{n} \in (a,b)\right) $$ Now the form in RHS can be simplified depending on moment constraints. I wonder if there are other methods...