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.
Vijayan N. Nair. Lawrence A. Shepp. Michael J. Klass. "On the Number of Crossings of Empirical Distribution Functions." Ann. Probab. 14 (3) 877 - 890, July, 1986. https://doi.org/10.1214/aop/1176992444