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

Arkadiusz Kasprzyk ; Władysław Szczotka

Applicationes Mathematicae, Tome 33 (2006), p. 129-143 / Harvested from The Polish Digital Mathematics Library

Résumé

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⌉.

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Publié le : 2006-01-01

Zbl 1110.60065

EUDML-ID : urn:eudml:doc:279453