A Nonlinear Renewal Theory with Applications to Sequential Analysis II

Self-Normalized Processes

“…Although the simple bounds (18.3) and related approximations are no longer applicable to the repeated GLR test, we can still use the likelihood ratio identity involving mixture of densities together with the pseudo-maximization method to analyze the error probabilities of the test. observations, Lai and Siegmund (1979) have derived asymptotic expansions for E(T α,β ) or E(Ñ), up to the o(1) term, by making use of nonlinear renewal theory; see Problem 18.4 for a sketch of the basic ideas. To obtain asymptotic approximations for the expected sample sizes of SPRTs for the general setting of dependent random variables, Lai (1981, p. 326) make use of (18.12) and uniform integrability after strengthening the a.s. convergence in (18.8) into r-quick convergence.…”

“…(18.55) Lai and Siegmund (1979) have extended Blackwell's renewal theorem (18.50) to U(x) in which S n is replaced by Z n =S n + ζ n in (18.49), where ζ n is slowly changing and satisfies some additional assumptions, including that ζ n converges in distribution to ζ . Lettingμ = ES 1 , they have used this result to show that (18.55) can be extended to (18.56) where T (b) = inf{n ≥ 1 : Z n ≥ b}.…”

Functional limit theorems for the maxima of perturbed random walk and divergent perpetuities in the M 1-topology

“…This is the setting of Theorem 1.3 in [11]. Plainly, condition (14) holds with ν 0 = ε (1/2,2,1) . Setting f 0 (t) = g(t) :…”

Biography of David O. Siegmund

“…Before 1985, his research concentrated on sequential analysis: the study of how data should be accumulated in an experimental situation. Siegmund's primary focus was on the design and analysis of sequential clinical trials that allow pharmaceutical workers to assess whether a new medicine is better or worse than an existing one (7)(8)(9)(10)(11)(12)(13).…”