On the distribution of tail array sums for strongly mixing stationary sequences
This paper studies the asymptotic distributions of ``tail array'' sums of the form \(\sum\psi_n(X_i-u_n)\) where \(\{X_i\}\) is a strongly mixing stationary sequence, \(\psi_n\) are real functions which are constants for negative arguments, \(\psi_n(x)=\psi_n(X_+)\), and \(\{u_n\}\) are levels with \(u_n \to\infty\). The random variables \(\sum 1_{(X_i-u_n>0)}\) and \(\sum(X_i-u_n)1_{( X_i-u_n>0)}\) are simple but important examples. Two cases of the asymptotic behaviour of \(\sum\psi_n(X_i-u_n)\) are considered: (1) compound Poisson convergence \((u_n\to\infty\) at a rapid rate defined by \(n(1-F(u_n))\to\tau< \infty)\) and (2) normal convergence \((u_n\to\infty\) more slowly, with \(n(1-F(u_n))\to\infty)\). A more detailed account is given for case (2). A final main result of this paper provides a simple approach to the verification of the Lindeberg condition for the important special cases of (approximately) exponential or (by a logarithmic transformation) regularly varying tails.
scientific article; zbMATH DE number 1319836
scientific article