### Projective stochastic equations and nonlinear long memory

#### Abstract

A projective moving average {Xt, tZ} 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.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1084-1105.

Dates
First available in Project Euclid: 12 December 2014

https://projecteuclid.org/euclid.aap/1418396244

Digital Object Identifier
doi:10.1239/aap/1418396244

Mathematical Reviews number (MathSciNet)
MR3290430

Zentralblatt MATH identifier
1305.60026

#### Citation

Grublytė, Ieva; Surgailis, Donatas. Projective stochastic equations and nonlinear long memory. Adv. in Appl. Probab. 46 (2014), no. 4, 1084--1105. doi:10.1239/aap/1418396244. https://projecteuclid.org/euclid.aap/1418396244

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