Beyond the tangent approximation and back to the Kolmogorov-Petrovski-Erdös test

Abstract

Let ψ(t) denote an increasing and continuously differentiable function. Let T=inf{t>0 | W(t)≧ψ(t)} denote the first exit time of the standard Brownian motion W(t) over ψ(t) with T=^∞ of the infimum is taken over the empty set. Let P(T>0)=1 and let p(t) denote the density of the distribution of T.