Asymptotics of Sequential Composite Hypothesis Testing Under Probabilistic Constraints

We consider the sequential composite binary hypothesis testing problem in which one of hypotheses is governed by a single distribution while other family distributions whose parameters belong to known set $\Gamma $ . would like design test decide effect. Under constraints that probabilities length test, stopping time, exceeds notation="LaTeX">$n$ are bounded certain threshold notation="LaTeX">$\epsilon , we obtain fundamental limits on asymptotic behavior as tends infinity. Assuming convex and compact set, all first-order error exponents for problem. also prove strong converse. Additionally, second-order under assumption alphabet observations notation="LaTeX">$\mathcal {X}$ finite. In proof asymptotics, main technical contribution derivation central limit-type result maximum an uncountable log-likelihood ratios suitable conditions. This may be independent interest. show some important statistical models satisfy