Let $F$ and $G$ be two continuous distribution functions that cross at a finite number of points $-\infty \leq t_1 < \cdots < t_k \leq \infty$. We study the limiting behavior of the number of times the empirical distribution function $G_n$ crosses $F$ and the number of times $G_n$ crosses $F_n$. It is shown that these variables can be represented, as $n \rightarrow \infty$, as the sum of $k$ independent geometric random variables whose distributions depend on $F$ and $G$ only through $F'(t_i)/G'(t_i), i = 1, \ldots, k$. The technique involves approximating $F_n(t)$ and $G_n(t)$ locally by Poisson processes and using renewal-theoretic arguments. The implication of the results to an algorithm for determining stochastic dominance in finance is discussed.
@article{1176992444,
author = {Nair, Vijayan N. and Shepp, Lawrence A. and Klass, Michael J.},
title = {On the Number of Crossings of Empirical Distribution Functions},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 877-890},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992444}
}
Nair, Vijayan N.; Shepp, Lawrence A.; Klass, Michael J. On the Number of Crossings of Empirical Distribution Functions. Ann. Probab., Tome 14 (1986) no. 4, pp. 877-890. http://gdmltest.u-ga.fr/item/1176992444/