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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.

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