Spectral analysis of multifractional LRD functional time series

Abstract

Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak-consistent parametric estimator of the long-memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005).

Hilbert space-valued fractionally integrated autoregressive moving average processes with long memory operators

“…In [1], the long memory part of the functional time series has a finite rank in the sense that it lives in a finite dimensional linear subspace. In [32], a class of long memory processes is introduced by imposing the spectral density operator to be of the form λ A M (λ) as the frequency λ → 0, with (A, M (λ)) λ∈(−π,π) a family of self-adjoint operators sharing the same decomposition operator (hence commuting), see Assumptions II and III in [32] (where ω refers to the frequency while λ refers to an element of σ(D)). In opposition to these works, the class of FIARMA processes introduced above neither impose a finite rank long memory, nor a particular commuting structure between a power law frequency behavior and a multiplicative bounded operator valued function.…”

“…In particular [32, Example 1] is a very specific subclass of FIARMA processes where the long memory operator and all the AR and MA operators are assumed to be self-adjoint and compact with the same eigenvectors, see [32,Eq. (3.16) and (3.17)].…”

“…Over the past several decades, the study of weakly stationary time series valued in a separable Hilbert space has been an active field of research. For example, functional ARMA processes were discussed in [3,33,22], a spectral theory was detailed in [27,26,34] and several estimation methods were studied in [18,19,17,20,23,2,24,9,1,32]. However, these references mainly focus on short memory processes.…”