Asymptotic optimality for one class of truncated sequential tests (Q1177969)

The author considers a class of stochastic processes of exponential type, which are homogeneous processes \((x_ t,\;t\geq 0)\) with independent increments satisfying \(P_ \theta(x_ 0=0)=1\) and such that for all \(x>0\), \(x\in E\), there exists \(r(t,x)>0\) and \(\psi(\theta)\) such that \[ P_ \theta(x_ t\in B)=\int_ B \exp(\theta x- t\psi(\theta))r(t,x)V_ t(dx) \] where \(B\) is any Borel subset of \(E\) and \(V_ t\) is a measure. After providing a definition of asymptotic optimality of the tests, the author introduces a class of truncated sequential tests and presents a series of lemmas and theorems showing the asymptotic optimality of the tests. Examples in the cases of Poisson and Wiener processes are given.