Compound Poisson approximation of the number of exceedances in Gaussian sequences
For a finite stationary sequence of Gaussian random variables \(y_1,y_2,\dots,y_n\) with \(\rho_k=Ey_1y_{k+1}\), \(k=2,\dots,n-1\), define \(W\) as the number of variables exceeding some given level \(a\). When some \(\rho_k\) are positive and large, the author proposes to use the compound Poisson distribution which provides better rate of convergence for approximation of the true distribution of \(W\) than the ordinary Poisson distribution. Thus, a short review of the local version of the Stein approximation method for compound Poisson distribution is given and this general theory is applied to the problem of exceedances of \(m\)-dependent Gaussian sequences. The upper bounds for total variation distance between \(W\) and compound Poisson variable are obtained in the case of a finite sequence and any positive \(a\) as well as the asymptotic bounds in the case, when \(n\) and \(a\) increase, while \(EW\) is fixed. Approximation using only two-term compound Poisson distribution is discussed and results of numerical calculations and simulations are presented.
scientific article; zbMATH DE number 1323314