Beijing Institute of Mathematical Sciences and Applications

We consider the problem of estimating the operator of a Hilbert-valued moving average process of order 1 (MAH(1)). Such processes, of the form Xₙ = εₙ + L(εₙ₋₁), arise naturally as infinite-dimensional generalizations of classical MA(1) models and provide a flexible framework for functional time series arising in insurance, finance, and other applied fields.
The central challenge is that, unlike the finite-dimensional setting, the operator L cannot be recovered by simple matrix inversion: the covariance operators C and D are compact and non-invertible on an infinite-dimensional Hilbert space. We address this by projecting the moment equation L²D* − LC + D = 0 onto a finite-dimensional subspace spanned by a common eigenvector basis of C and D, yielding a tractable system of equations for the eigenvalues of L.
We present the theoretical results, establishing the almost sure convergence of the empirical eigenvalue estimators of C and D, the convergence rate of the eigenvalue estimators of L, the convergence of the operator estimator L̂, and a uniform convergence result for the projection estimator. The method is illustrated on an application to insurance turnover data.