The Annals of Mathematical Statistics

On the Number of Solutions of Systems of Random Equations

David R. Brillinger

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

Citation

Brillinger, David R. On the Number of Solutions of Systems of Random Equations. Ann. Math. Statist. 43 (1972), no. 2, 534--540. doi:10.1214/aoms/1177692634. https://projecteuclid.org/euclid.aoms/1177692634


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