Weak convergence of the function-indexed integrated periodogram for infinite variance processes

Abstract

In this paper we study the weak convergence of the integrated periodogram indexed by classes of functions for linear and stochastic volatility processes with symmetric alpha-stable noise. Under suitable summability conditions on the series of the Fourier coefficients of the index functions we show that the weak limits constitute alpha-stable processes which have representation as infinite Fourier series with iid alpha-stable coefficients. The cases alpha in (0,1) and alpha in [1,2) are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case alpha in (0,1), entropy conditions are needed for alpha in [1,2) to ensure the tightness of the sequence of integrated periodograms indexed by functions.

The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.