Combinatorial devices for sequential analysis

Abstract

For Brownian motion, escape probabilities over curved boundaries have been studied in some detail. Since sequential analysis is not concerned with really large samples, the approximation by the Brownian motion is questionable; neither the central limit theorem nor the renewal theorem regulating the overshoot may be appropriate tools. The present paper shows that combinatorial results developed in fluctuation theory have some bearing on the calculation of escape probabilities. The main result is a kind of tangential approximation for random walks crossing a curved boundary.