The author studies the convergence rates of probabilities of type I and type II errors for a sequence of Neyman-Pearson tests \({\hat \phi}=({\hat \phi}^ n)\) in the case where, for each \(n\geq 1\), a filtering \({\mathcal F}=({\mathcal F}^ n_ t)_{t\in R_+}\), \({\mathcal F}^ n=\bigvee_{t\in R_+}{\mathcal F}^ n_ t\), is specified on a measurable space \((\Omega^ n,{\mathcal F}^ n)\), which satisfies the usual conditions with respect to a measure \(Q^ n\) and where the distributions \(P^ n\) and \(\tilde P^ n\) under the related simple hypotheses are assumed to be probability distributions of random processes in the class of semi-martingales.
The limit theorems obtained in this paper strengthen and generalize the corresponding results obtained by Yu. N. Lin’kov [Theory Probab. Math. Stat. 35, 65-73 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 60-69 (1986; Zbl 0632.62022), and Teor. Sluchajnykh Protsessov 14, 48-55 (1986)].