On the Asymptotic Distribution of a Certain Functional of the Wiener Process
Wyner, A. D.
Ann. Math. Statist., Tome 40 (1969) no. 6, p. 1409-1418 / Harvested from Project Euclid
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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 $N \rightarrow \infty$.
Publié le : 1969-08-14
Classification:  
@article{1177697512,
     author = {Wyner, A. D.},
     title = {On the Asymptotic Distribution of a Certain Functional of the Wiener Process},
     journal = {Ann. Math. Statist.},
     volume = {40},
     number = {6},
     year = {1969},
     pages = { 1409-1418},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177697512}
}

Wyner, A. D. On the Asymptotic Distribution of a Certain Functional of the Wiener Process. Ann. Math. Statist., Tome 40 (1969) no. 6, pp.  1409-1418. http://gdmltest.u-ga.fr/item/1177697512/