On Seasonal Autoregressive Processes Inference

In the study of several real time series and other major fluctuations such as the trend,the cycle and the noise, the presence of seasonal fluctuations is one of the most important issues.
The investigation has an established practice provided that seasonal variations have been
regarded as a disruptive element and then must be eliminated. However, these fluctuations are
an integral part that must be studied in order to evaluate and forecast the studied model. A
well-known practice for modeling seasonal data is to utilise an autoregressive model that is able
to handle the presence of the seasonal patterns in the data. Autoregressive models are a kind
of time-series model that utilise lagged values of the target variable to make predictions about
future values. Notice that despite the fact that these data are obtainable in practice as sequences of discrete
observed values, they are basically approached as functions. Functional autoregressive models are well known for the analysis of time series
analysis. However, basic formulation is not suitable for investigating the seasonal behaviour
in functional time series data. Hence, we introduce seasonal functional autoregressive processes to model time series. For the autoregressive process of order one, we provide
conditions of stationarity and formulate limit theorems, and supply methods of estimation
and prediction. The worthiness of these models is displayed via algorithmic investigations.