Covariance structure of wide-sense Markov processes of order $k\geq 1$
Tom 33 / 2006
Streszczenie
A notion of a wide-sense Markov process $\{X_t\}$ of order $k\geq 1,$ $\{X_t\} \sim {\rm WM}(k),$ is introduced as a direct generalization of Doob's notion of wide-sense Markov process (of order $k=1$ in our terminology). A base for investigation of the covariance structure of $\{X_t\}$ is the $k$-dimensional process $\{x_t=(X_{t-k+1},\dots, X_t)\}.$ The covariance structure of $\{X_t\}\sim {\rm WM}(k)$ is considered in the general case and in the periodic case. In the general case it is shown that $\{X_t\}\sim {\rm WM}(k)$ iff $\{x_t\}$ is a $k$-dimensional ${\rm WM}(1)$ process and iff the covariance function of $\{x_t\}$ has the triangular property. Moreover, an analogue of Borisov's theorem is proved for $\{x_t\}.$ In the periodic case, with period $d>1,$ it is shown that Gladyshev's process $\{Y_t=(X_{(t-1)d+1}, \dots, X_{td})\}$ is a $d$-dimensional ${\rm AR}(p)$ process with $p= \lceil k/d \rceil .$
