Non-Stationary Stochastic Processes Estimation

9 Interpolation of processes with periodically correlated increments observed with or without noise

In this chapter, we deal with the interpolation problem for stochastic processes with periodically stationary dth increments observed with a periodically stationary stochastic noise process.

By the interpolation problem we understand the problem of the mean square optimal linear estimation of the functional

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

which depends on the unknown values of the stochastic process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ with periodically correlated dth increments. Estimates are based on observations of the process $\mathit{\zeta }\mathrm{(}\mathit{t}\mathrm{)}=\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}+\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ at points $\mathit{t}\in \mathbb{R}\setminus [0;\mathrm{(}\mathit{N}+1\mathrm{)}\mathit{T}]$, where $\mathit{\eta }\mathrm{(}\mathit{t}\mathrm{)}$ is an uncorrelated with the process $\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)}$ periodically stationary stochastic process.

9.1 Hilbert ...