The Annals of Mathematical Statistics

On the Number of Solutions of Systems of Random Equations

David R. Brillinger

Abstract

Let $\{f(x, \omega); x \in R^n, \omega \in \Omega \}$ be an $n$ vector-valued stochastic process defined over a probability space $(\Omega, \mathscr{A}, \mu)$. Let $N(f \mid A, y)$ denote the number of elements in the set $A \cap f^{-1}(y)$, that is the number of distinct solutions of the system of equations $f(x, \omega) = y$ for $x, y \in R^n$. We develop expressions for $E\{N(f \mid A, y)\}$ and certain higher-order moments of $N(f \mid A, y)$ under regularity conditions.

Article information

Source
Ann. Math. Statist., Volume 43, Number 2 (1972), 534-540.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177692634

Digital Object Identifier
doi:10.1214/aoms/1177692634

Mathematical Reviews number (MathSciNet)
MR300329

Zentralblatt MATH identifier
0238.60040

JSTOR