## The Annals of Mathematical Statistics

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

A. D. Wyner

#### 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 $N \rightarrow \infty$.

#### Article information

Source
Ann. Math. Statist., Volume 40, Number 4 (1969), 1409-1418.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177697512

Digital Object Identifier
doi:10.1214/aoms/1177697512

Mathematical Reviews number (MathSciNet)
MR240875

Zentralblatt MATH identifier
0185.44703

JSTOR