Asymptotic Comparisons of Functionals of Brownian Motion and Random Walk

In this paper we make comparisons involving stopping times $\tau$ of a process $X$ and the maximal function $X^\ast_\tau$ of that process, where $X$ is either Brownian motion or random walk. In particular, we give conditions implying that $P(X^\ast_\tau > \lambda) \approx P(\tau^{1/2} > \lambda)$ in the sense of a two-sided inequality holding. We show that if, for all large $\lambda$ there exist constants $\beta > 1$ and $\gamma > 0$ satisfying $$0 < P(\tau^{1/2} > \lambda) \leq \gamma P(\tau^{1/2} > \beta\lambda),$$ and if $X$ is a one-dimensional Brownian motion, then $P(X^\ast_\tau > \lambda) \approx P(\tau^{1/2} > \lambda)$. An analogous result is given for $n$-dimensional Brownian motion $(n \geq 3)$. We also consider a similar result for one-sided maximal functions of local martingales. Finally, we look at a random walk $X$, where $X_n = x_1 + x_2 + \cdots + x_n$, and give two different sets of conditions on $\tau$ and the $x_i$'s under which the result $P(\tau^{1/2} > \lambda) \approx P(X^\ast_\tau > \lambda)$ is true.