This is a research report about my work on sequential statistic~ during 1980 - 1984. Two themes are treated which are closely related to each other and to the law of the iterated logarithm:· I) curved boundary first passage distributions of Brownian motion, 11) optimal properties of sequential tests with parabolic and nearly parabolic boundaries. In the first chapter I discuss the tangent approximation for Brownianmotion as a global approximation device. This is an extension of Strassen' s approach to t'he law of the iterated logarithm which connects results of fluctuation theory of Brownian motion with classical methods of sequential statistics. In the second chapter I make use of these connections and derive optimal properties of tests of power one and repeated significance tests for the simpiest model of sequential statistics, the Brownian motion with unknown drift. To both topics:there under1ies an asymptotic approach which is closely linked to large deviation theory: the stopping boundaries recede to infinity. This is a well-known approach in sequential stötistics which is extensively discussed in Siegmund's recent book ·Sequential Analysis". This approach also leads to some new insights about the law of the iterated logarithm (LIL). Although the LIL has been studied for nearly seventy years the belief is still common that it applies only for large sampIe sizes which can never be obser ved in practice.

Inhalt
I: Curved Boundary First Passage Distributions Of Brownian Motion.- 1. The general method of images for the diffusion equation.- 2. The method of weighted likelihood functions.- 3. From the method of images to the tangent approximation.- 4. The tangent approximation.- 5. Beyond the tangent approximation and back to the KolmogorovPetrovski-Erdös test.- II: Optimal Properties of Sequential Tests with Parabolic and Nearly Parabolic Boundaries.- 1. Bayes tests of power one.- 2. An application of the tangent approximation: a heuristic derivation of the shape of Bayes tests of power one.- 3. Construction of tests of power one from smooth priors and the law of the iterated logarithm for posterior distributions.- 4. Exact results about the shape.- 5. An optimal property of the repeated significance test.- References.