Functional Periodic ARMA Processes

Periodicity is a common feature of time series. For finite-dimensional data, periodic autoregressive moving average (ARMA) models have been extensively studied. In functional time series analysis, AR models have been extended to incorporate periodicity, but existing approaches remain incomplete and do not cover the ARMA setting. This paper develops a rigorous theoretical framework for functional periodic ARMA (fPARMA) processes in general separable Hilbert spaces. The proposed model class accommodates periodically varying dependence structures. We derive sufficient conditions for periodic stationarity, the existence of finite moments, and weak dependence. Moreover, we study Yule-Walker-type estimators for the fPAR operators and, in a specific setting, estimators for the fPARMA operators, and establish convergence rates under Sobolev-type regularity assumptions.