Boundary Crossing Probabilities for the Wiener Process and Sample Sums

Let W (t) denote a standard Wiener process for 0 t 0 (or for some t > 0) for a certain class of functions g (t), including functions which are (2t t) ^1{2} as t. We also prove an invariance theorem which states that this probability is the limit as m of the probability that Sₙ m^1{2}g (n/m) for some n m (or for some n 1), where Sₙ is the nth partial sum of any sequence x₁, x₂, of independent and identically distributed (i. i. d. ) random variables with mean 0 and variance 1. The main results were announced in 19. Some aspects of the invariance theorem were considered independently by Muller 14, who also studied the rate of convergence to the limiting distribution. Statistical applications of these ideas are indicated in 3 and 18. In Section 2 we state the general theorems and give several examples. Sections 3-5 are devoted to the proof of these results. In Section 6 we indicate the applicability of our methods to stochastic processes other than the Wiener process. Of particular interest in this regard is the analogue of Theorem 1 for Bessel diffusion processes. Section 7 raises questions which will be treated in a subsequent paper.