On the number of crossings of empirical distribution functions. (English) Zbl 0593.60047

Nair, Vijayan N.; Shepp, Lawrence A.; Klass, Michael J.

Ann. Probab. 14, 877-890 (1986).

Summary: Let F and G be two continuous distribution functions that cross at a finite number of points \(-\infty \leq t_ 1<...<t_ k\leq \infty\). 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 \(n\to \infty\), as the sum of k independent geometric random variables whose distributions depend on F and G only through \(F'(t_ i)/G'(t_ i)\), \(i=1,...,k.\) 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.

MSC:

60G17	Sample path properties
60E05	Probability distributions: general theory

Keywords:

asymptotic distribution; boundary crossing probability; Wiener-Hopf technique; empirical distribution; renewal-theoretic arguments