Non-Stationary Stochastic Processes Estimation

7 Extrapolation of processes with periodically correlated increments

In this chapter, we deal with the extrapolation problem for stochastic processes with periodically correlated dth increments. More precisely, we consider the problem of the mean square optimal linear estimation of the functionals

$\begin{array}{c}\mathit{A}\mathit{\xi }=\underset{0}{\overset{\infty }{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}{\mathit{A}}_{\mathit{N}\mathit{T}}\mathit{\xi }=\underset{0}{\overset{\mathrm{(}\mathit{N}+1\mathrm{)}\mathit{T}}{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\mathrm{,}\end{array}$

which depend on the unknown values of a stochastic process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ with periodically correlated dth increments. Estimates are based on observations of the process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ at points .

7.1 Generated stationary increment sequence representation of the functional Aξ

Lemma 7.1.1 (Luz and Moklyachuk [115]).

Any linear functional

$\begin{array}{c}\mathit{A}\mathit{\xi }=\underset{0}{\overset{\infty }{\int }}\mathit{a}\mathrm{(}\mathit{t}\mathrm{)}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}\mathit{d}\mathit{t}\end{array}$

allows a representation

$\begin{array}{c}\mathit{A}\mathit{\xi }=\mathit{B}\mathit{\xi }-\mathit{V}\mathit{\xi }\mathrm{,}\end{array}$

where

$\begin{array}{c}\mathit{B}\mathit{\xi }=\underset{0}{\overset{\infty }{\int }}{\mathit{b}}^{\mathit{\tau }}\mathrm{(}\mathit{t}\end{array}$