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.
Citation
David R. Brillinger. "On the Number of Solutions of Systems of Random Equations." Ann. Math. Statist. 43 (2) 534 - 540, April, 1972. https://doi.org/10.1214/aoms/1177692634
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