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Projective Stochastic Equations and Nonlinear Long Memory

Part of: Stochastic analysis Stochastic processes Limit theorems

Published online by Cambridge University Press:  22 February 2016

Ieva Grublytė  and
Donatas Surgailis
Ieva Grublytė*
Affiliation:
Vilnius University
Donatas Surgailis*
Affiliation:
Vilnius University
*
∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania.
∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, LT-08663 Vilnius, Lithuania.
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Abstract

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A projective moving average {Xt, t ∈ ℤ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on ‘intermediate’ lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Keywords

Projective stochastic equationlong memoryLARCH modelBernoulli shiftinvariance principle

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60F17: Functional limit theorems; invariance principles 60H25: Random operators and equations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1084 - 1105
Copyright
© Applied Probability Trust 

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