Projective stochastic equations and nonlinear long memory

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.