On the Number of Crossings of Empirical Distribution Functions

Abstract

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.