Setup: I have a finite family of independent $X_t$ that will be sampled sequentially based on the fact that at each observation we will proceed with a two-tailed test against our hypothesis. Let $S_t= \sum_{i=0}^{t} X_t$, then the null hypothesis is $H_0: S_t > \Delta$ and the alternative is $H_1: S_t < \Delta$. Both hypothesis are composite and inexact. I am mainly interested in the practical aspect. I do not have an infinite stream of observations my goal is to minimize the stopping time for a given confidence level.
What I know: From what I have gathered, searching for references on Google Scholar, the closest bet was the generalized sequential probability ratio test (GSPRT). However, the papers often only tackle specific families of distributions and it is often the case that these distributions are not heavy-tailed which is likely to be my case or the papers do not tackle composite hypothesis. Furthermore, the assumption that a normal distribution can approximate $S_t$ is likely to be inexact a better approximation would be that the $X_t$ follow Cauchy distributions therefore $S_t$ also does but I would like to have a non parametric solution.
What I would like to know: I am interested in anything that could help me tackle my problem. My main interest is in Type I error but if there's anything relatively close for other type of errors, I also am interested. If you have any references or idea, I would be interested in, please do share!
Perhaps my best bet would be to have a parametric solution in which I specify the distribution of $S_t$?
Thanks