Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Seque

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.