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## Applicationes Mathematicae

2006 | 33 | 2 | 129-143

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

EN

### Abstrakty

EN
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⌉.

129-143

wydano
2006

### Twórcy

autor
• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
autor
• Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland