Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, and Władysław Szczotka. "Covariance structure of wide-sense Markov processes of order k ≥ 1." Applicationes Mathematicae 33.2 (2006): 129-143. <http://eudml.org/doc/279453>.

@article{ArkadiuszKasprzyk2006,
abstract = {A notion of a wide-sense Markov process $\{X_t\}$ of order k ≥ 1, $\{X_t\} ∼ WM(k)$, 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 $\{x_t = (X_\{t-k+1\},...,X_t)\}$. The covariance structure of $\{X_t\} ∼ WM(k)$ is considered in the general case and in the periodic case. In the general case it is shown that $\{X_t\} ∼ WM(k)$ iff $\{x_t\}$ is a k-dimensional 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 $\{Y_t = (X_\{(t-1)d+1\},...,X_\{td\})\}$ is a d-dimensional AR(p) process with p = ⌈k/d⌉.},
author = {Arkadiusz Kasprzyk, Władysław Szczotka},
journal = {Applicationes Mathematicae},
language = {eng},
number = {2},
pages = {129-143},
title = {Covariance structure of wide-sense Markov processes of order k ≥ 1},
url = {http://eudml.org/doc/279453},
volume = {33},
year = {2006},
}

TY - JOUR
AU - Arkadiusz Kasprzyk
AU - Władysław Szczotka
TI - Covariance structure of wide-sense Markov processes of order k ≥ 1
JO - Applicationes Mathematicae
PY - 2006
VL - 33
IS - 2
SP - 129
EP - 143
AB - A notion of a wide-sense Markov process ${X_t}$ of order k ≥ 1, ${X_t} ∼ WM(k)$, 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 ${x_t = (X_{t-k+1},...,X_t)}$. The covariance structure of ${X_t} ∼ WM(k)$ is considered in the general case and in the periodic case. In the general case it is shown that ${X_t} ∼ WM(k)$ iff ${x_t}$ is a k-dimensional 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 ${Y_t = (X_{(t-1)d+1},...,X_{td})}$ is a d-dimensional AR(p) process with p = ⌈k/d⌉.
LA - eng
UR - http://eudml.org/doc/279453
ER -