Poisson approximation of the number of exceedances of a discrete-time x2-process

Consider a discrete-time x2-process, i.e. a process defined as the sum of squares of independent and identically distributed Gaussian processes. Count the number of values that exceed a certain level. Let this level and the number of time points considered increase simultaneously so that the expected number of points above the level remains fixed. It is shown that the number of exceeding points converges to a Poisson distribution if the dependence in the underlying Gaussian processes is not too strong. By using the coupling approach of the Stein-Chen method, both limit theorems and rates of convergence are obtained.