# Is AR(1) a Markov process?

Is AR(1) process such as
$y_t=\rho y_{t-1}+\varepsilon_t$ a Markov process?

If it is, then VAR(1) is the vector version of Markov process?

The following result holds: If $\epsilon_1, \epsilon_2, \ldots$ are independent taking values in $E$ and $f_1, f_2, \ldots$ are functions $f_n: F \times E \to F$ then with $X_n$ defined recursively as

the process $(X_n)_{n \geq 0}$ in $F$ is a Markov process starting at $x_0$. The process is time-homogeneous if the $\epsilon$‘s are identically distributed and all the $f$-functions are identical.

The AR(1) and VAR(1) are both processes given in this form with

Thus they are homogeneous Markov processes if the $\epsilon$‘s are i.i.d.

Technically, the spaces $E$ and $F$ need a measurable structure and the $f$-functions must be measurable. It is quite interesting that a converse result holds if the space $F$ is a Borel space. For any Markov process $(X_n)_{n \geq 0}$ on a Borel space $F$ there are i.i.d. uniform random variables $\epsilon_1, \epsilon_2, \ldots$ in $[0,1]$ and functions $f_n : F \times [0, 1] \to F$ such that with probability one

See Proposition 8.6 in Kallenberg, Foundations of Modern Probability.