On Stopping Times for $n$ Dimensional Brownian Motion

Abstract

Let $\bar{X}(t) = (X_1(t),\cdots, X_n(t))$ be standard $n$ dimensional Brownian motion. Results of the following nature are proved. If $\tau$ is a stopping time for $\bar{X}(t)$ then $|\bar{X}(\tau)|$ and $(n\tau)^{\frac{1}{2}}$ are relatively close in $L^p$ if $n$ is large. Also, if $n$ is large most of the moments $EX_i(\tau)^k, i = 1,2,\cdots, n$, are about what they would be if $\bar{X}(t)$ were independent of $\tau$.