On the Asymptotic Distribution of a Certain Functional of the Wiener Process

Abstract

Let the random variable $Y_N$ be defined by $Y_N = \sum^N_{k=1} W^2(k)/k^2,$ where $W(t)$ is the Wiener process, the Gaussian random process with mean zero and covariance $EW(s)W(t) = \min (s, t)$. Note that $EY_N \sim \log N$. We show that for $a > 1$ $\operatorname{Pr}\lbrack Y_N \geqq a \log N \rbrack = N^{-(8a)^{-1}(a-1)^2(1+\epsilon_N)},$ where $\epsilon_N \rightarrow 0$ as $N \rightarrow \infty$.