Projective Stochastic Equations and Nonlinear Long Memory. (December 2014)

• Record Type: Journal Article
• Authors: Grublytė, Ieva; Surgailis, Donatas
• Is Part Of: Advances in applied probability. Volume 46:Number 4(2014)
• Page Start: 1084
• Page End: 1105
• Publication Date: 2014-12
• Subjects: Projective stochastic equation, -- long memory, -- LARCH model, -- Bernoulli shift, -- invariance principle; 60G10, -- 60F17, -- 60H25; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1239/aap/1418396244
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract : A projective moving average { X t, 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 X t on 'intermediate' lagged innovation subspaces with given coefficients α i and β i, 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 X t . We show that, under certain conditions on Q, α i, and β i, j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.