Poisson Approximation Using the Stein-Chen Method and Coupling: Number ofExceedances of a Discrete-Time-chi(2)-Process

摘要

Consider a discrete-time chi squared process, i.e. a process defined as the sumof squares of independent and identically distributed normal processes. Count the number of values that exceeds a certain level. Let this level grow together with the number of time-points considered so that the expected number of points above the level remains fixed. It is shown that the number of exceeding points converges in distribution to a Poisson distribution, if the dependence in the underlying normal processes is not too strong. By using the coupling approach of the Stein-Chen method, both limit theorems and rates of convergence can be obtained. Finally the convergence is illustrated by numerical examples on ARMA-processes.