On stationary Markov chains and independent random variables (Q908588)

The Rosenblatt representation theorem for a stationary, irreducible, aperiodic Markov chain \((X_ n:\) \(n\in Z)\) on a countable state space E asserts the existence of a measurable function \(g:I\to E,\) such that the sequence \((g(U_{n-1},U_{n-2},...)\), \(n\in Z)\) is law equivalent to \((X_ n:\) \(n\in Z)\). (I is the classic unit interval and \(U_ n\) is the n-th projection of \(I^ Z\) into I). Two new proofs are given in the article. Indeed, if \(\mu\) is a probability distribution on a nice space E \((=\{1,2,...\}\) for the article) it can be represented by a measurable function \(\xi:I\to E,\) such that the distribution of \(\xi\) is \(\mu\) :\(\xi\) \(\sim \mu\). If P is a Markov kernel on E, it can be represented by a measurable function \(f:\quad E\times I\to E,\) such that f(x,\(\cdot)\sim P(x,\cdot)\). Define now \(X_ 0(x)\equiv x\), \(X_ 1\equiv f\), and recursively \[ X_{n+1}(x,U_ 0,U_ 1,...,U_ n)=f(X_ n(x,U_ 0,U_ 1),U_ n). \] For any r.v. \(X'_ 0: \Omega '_ 0\to E,\) the sequence defined by \[ X'_ n,\quad n\in Z_+=\{0,1,...\},\quad X'_ n\equiv X_ n(X'_ 0,U_ 0,...,U_{n-1}) \] is a Markov chain with transition kernel P, on the product space \(\Omega '_ 0\times I\). If \(X'_ 0\) is a distribution stationary for P, \(([U_ n,X'_ n]\), \(n\in Z)\) is a stationary sequence, and can be extended to negative n, by enlarging the ground space if necessary. For the irreducible, aperiodic and positive recurrent chain P, it is proved that \[ \lim_{n\to \infty}X_ n(*,U_{-n},...,U_{- 1})=g(U_{-1},U_{-2},...) \] where * is any fixed point of E.