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Aggregation of a random-coefficient ar(1) process with infinite variance and idiosyncratic innovations

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Donata Puplinskaitė  and
Donatas Surgailis
Donata Puplinskaitė*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics, Vilnius
Donatas Surgailis*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania. Email address: donata.puplinskaite@mif.stud.vu.lt
∗∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania. Email address: sdonatas@ktl.mii.lt
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Abstract

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Contemporaneous aggregation of N independent copies of a random-coefficient AR(1) process with random coefficient a ∈ (−1, 1) and independent and identically distributed innovations belonging to the domain of attraction of an α-stable law (0 < α < 2) is discussed. We show that, under the normalization N 1/α, the limit aggregate exists, in the sense of weak convergence of finite-dimensional distributions, and is a mixed stable moving average as studied in Surgailis, Rosiński, Mandrekar and Cambanis (1993). We focus on the case where the slope coefficient a has probability density vanishing regularly at a = 1 with exponent b ∈ (0, α − 1) for α ∈ (1, 2). We show that in this case, the limit aggregate {X̅ t } exhibits long memory. In particular, for {X̅ t }, we investigate the decay of the codifference, the limit of partial sums, and the long-range dependence (sample Allen variance) property of Heyde and Yang (1997).

Keywords

Aggregationrandom-coefficient AR(1) processinfinite variancemixed stable moving averagecodifferenceself-similar processlong memory

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G18: Self-similar processes 60G52: Stable processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 509 - 527
Copyright
Copyright © Applied Probability Trust 2010 

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