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- W2034144023 abstract "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 $$0 < P(tau^{1/2} > lambda) leq gamma P(tau^{1/2} > betalambda),$$ 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." @default.
- W2034144023 created "2016-06-24" @default.
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- W2034144023 date "1980-12-01" @default.
- W2034144023 modified "2023-09-23" @default.
- W2034144023 title "Asymptotic Comparisons of Functionals of Brownian Motion and Random Walk" @default.
- W2034144023 doi "https://doi.org/10.1214/aop/1176994574" @default.
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