Stationary Increments of Discrete Time Stochastic Processes: Spectral Representation

This chapter presents a brief review of the spectral theory of stochastic sequences with stationary nth increments. The spectral function and spectral density of the stochastic sequence are used with stationary increments to refer to the spectral function and the spectral density of the corresponding stationary increment sequence. Stochastic sequences with stationary increments allow people to describe a wide class of time series models, which are exploited in econometrics and financial time series theory. In particular, these are autoregression-moving-average sequences, seasonal time series, and integrated and cointegrated sequences. The chapter also presents a brief overview of definitions and properties of these sequences. The spectral theory of stationary increment sequences can be easily generalized to the increments with different steps.