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Can, Sami Umut; Mikosch, Thomas; Samorodnitsky, Gennady

Weak convergence of the function-indexed integrated periodogram for infinite variance processes. (English) Zbl 1207.62173

Bernoulli 16, No. 4, 995-1015 (2010).
Summary: 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.

Cited in 4 Documents

MSC:

62M15 Inference from stochastic processes and spectral analysis
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions

Keywords:

asymptotic theory; empirical spectral distribution; entropy; integrated periodogram; linear process; random quadratic form; spectral analysis; stable process; time series; weighted integrated periodogram
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