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Projective stochastic equations and nonlinear long memory. (English) Zbl 1305.60026
Summary: A projective moving average \(\{X_t: t\in \mathbb{Z}\}\) 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 \(\alpha_i\) and \(\beta_{i,j}\). The class of such equations includes the 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 \(Q\), \(\alpha_i\) and \(\beta_{i,j}\), this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

60G10 Stationary stochastic processes
60F17 Functional limit theorems; invariance principles
60H25 Random operators and equations (aspects of stochastic analysis)
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