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Strictly stationary solutions of ARMA equations with fractional noise
Author(s) -
Vollenbröker Bernd
Publication year - 2012
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2012.00788.x
Subject(s) - mathematics , noise (video) , a priori and a posteriori , moment (physics) , sequence (biology) , autoregressive–moving average model , moving average , series (stratigraphy) , stationary sequence , mathematical analysis , stochastic process , statistics , autoregressive model , computer science , paleontology , philosophy , physics , epistemology , classical mechanics , artificial intelligence , biology , image (mathematics) , genetics
We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations with fractional noise. Here, the underlying noise sequence of the fractional noise is assumed to be i.i.d. but no a priori moment assumptions are made. We also characterize for which i.i.d. driving noise sequences the series defining fractional noise converges almost surely. In the proofs, we use growth estimates for the moments of random walks developed by Manstavičius (1982) and techniques related to those of Brockwell and Lindner (2010) for the existence of strictly stationary ARMA processes with i.i.d. noise.