Usually when we define a Markov process of order $p$ for a univariate time-series $\{X_t\in\mathbb{R},t=1,2,\cdots\}$, the definition is presented as follows

\begin{equation} P(X_t\leq x_t\mid x_1,\cdots,x_{t-1})=P(X_t\leq x_{t}\mid x_{t-p},\cdots,x_{t-1}) \end{equation} however, let us now consider a vector process $\{Z_t=(x_t,y_t)'\in\mathbb{R}^2,t=1,2,\cdots\}$. Would the correct way to present it be \begin{equation} Z_t\mid Z_{1},\cdots,Z_{t-1}\sim Z_t\mid Z_{t-p},\cdots,Z_{t-1} \end{equation} My intuition is that presenting it as we did for the univariate cse would be incorrect. However, I saw a paper where it had shown the process in that manner and now I am confused.


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