On the mean estimation of variance for an asymptotically stationary stochastic process with bounded duration (Q1282553)

A stable linear stochastic dynamic system \[ \dot x(t)= Ax(t)+ B\xi(t),\quad t_0\leq t\leq t_0+ T,\quad x(t_0)= x_0\tag{1} \] is considered. Here \(x(t)\in \mathbb{R}^n\), \(A\) is a nonsingular matrix, \(\xi(t)\) is \(r\)-dimensional white noise, \(B\) is an \(n\times r\) matrix of distribution of control, and \(x_0\) is a random vector of initial states with \(Ex_0= 0\). By a single realization of the random process on the interval \([t_0,t_0+ T]\), the empirical estimation of the covariance matrix \(E[x(t)x^T(t)]\) is constructed, \[ \widehat D(t_0, T)= T^{-1}\cdot \int^{t_0+ T}_{t_0} [x(t)-\widehat m(t_0, T)]\cdot [x(t)-\widehat m(t_0, T)]^T dt, \] with \[ \widehat m(t_0, T)= T^{-1}\cdot \int^{t_0+ T}_{t_0} x(\lambda) d\lambda. \] An expression for the mean value of \(\widehat D(t_0, T)\) is derived for the process (1) in its nonstationary state. For the stationary state, this was done by \textit{A. N. Tikhonov} and \textit{V. N. Kharisov} [Statistical analysis and synthesis of radiotechnical devices and systems (Russian), Moscow: Radio i Svyaz' (1991)]. Two examples are given: a) the problem of the sliding average in time for a one-dimensional casual asymptotically stationary process, and b) estimation of the variation of the relief heights in foothill regions.