On the convergence of discrete schemes to continuous ones in some problems of sequential analysis

Two problems of sequential analysis are considered in the paper: the problem of «disorder» and the problem of sequential testing statistical hypotheses.
Let the observations at moments $k\Delta$ ($k=0,1,\dots,T/\Delta$), $\Delta\to 0$, are available, while the hypotheses considered get closer to each other. It is shown that the statistics $\pi_{\Delta}(t)$ and $\varphi_{\Delta}(t)$ converge to the diffusion processes $\pi(t)$ and $\varphi(t)$ (Lemmas 2–4) as $\Delta\to 0$. Conditions are also given (Theorems 2, 3) under which the convergence of the average lag time in the problem of «disorder» and the convergence of $\mathbf M_0\tau^{\Delta}$ and $\mathbf M_1\tau^{\Delta}$ in the problem of sequential testing statistical hypothesis follows from the convergence of these statistics.