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

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

2010-11-23

arxiv.org/abs/1011.5062v1

Bernoulli 2010, Vol. 16, No. 4, 995-1015

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ253 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ253

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $\alpha$-stable innovations. 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 representations as infinite Fourier series with i.i.d. $\alpha$-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.

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