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Mixed‐Norm Spaces and Prediction of S α S Moving Averages
Author(s) -
Cheng Raymond,
Harris Charles B.
Publication year - 2015
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/jtsa.12134
Subject(s) - mathematics , moving average , reciprocal , norm (philosophy) , moving average model , autoregressive model , autoregressive–moving average model , pure mathematics , autoregressive integrated moving average , mathematical analysis , statistics , time series , philosophy , linguistics , political science , law
Suppose thatZ kk = − ∞ ∞is an i.i.d. symmetric α ‐stable noise, 1 < α < 2, and consider the moving average processX kk = − ∞ ∞given byX k = ∑ j = 0 ∞a jZ k − j . Conditions are obtained for the convergence rate of the moving average series, as well as that of the inverted (autoregressive) representationZ k = ∑ j = 0 ∞c jX k − j . These conditions are expressed in terms of the associated function F ( z ) = ∑ j = 0 ∞a jz jand its reciprocal belonging to certain mixed‐norm spaces of functions on the open unit disc. Properties of these spaces are explored. Criteria are also derived for the rate of mixing in a certain sense.