Is AR(1) a Markov process?

Is AR(1) process such as
yt=ρyt1+εt a Markov process?

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

Answer

The following result holds: If ϵ1,ϵ2, are independent taking values in E and f1,f2, are functions fn:F×EF then with Xn defined recursively as

Xn=fn(Xn1,ϵn),X0=x0F

the process (Xn)n0 in F is a Markov process starting at x0. The process is time-homogeneous if the ϵ‘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

fn(x,ϵ)=ρx+ϵ.

Thus they are homogeneous Markov processes if the ϵ‘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 (Xn)n0 on a Borel space F there are i.i.d. uniform random variables ϵ1,ϵ2, in [0,1] and functions fn:F×[0,1]F such that with probability one
Xn=fn(Xn1,ϵn).
See Proposition 8.6 in Kallenberg, Foundations of Modern Probability.

Attribution
Source : Link , Question Author : Flying pig , Answer Author : NRH

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