A finite form of de Finetti's theorem for stationary Markov exchangeability
A finite length stochastic process on a finite state space C is said to be (i) Markov exchangeable, if realisations having the same number of i to j transitions for every pair of states i,j\(\in C\) are equiprobable, (ii) a mixture of Markov chains, if its distribution is a mixture of Markovian distributions. The main result of this paper asserts that if \(X_ 1,X_ 2,...,X_ n\) is stationary and Markov exchangeable, then there exists a process \(Y_ 1,Y_ 2,...,Y_ n\) on the same state space, which is a mixture of Markov chains such that the variation norm distance between the joint distribution of \(X_ 1,...,X_ k\) and the joint distribution of \(Y_ 1,...,Y_ k\) is no greater than \[ 2[c^ 2+c+1]\frac{k-1}{n-1}[1+\log (n-2)]\quad (k\leq n), \] where c is the cardinality of the state space. This is analogous to a result of \textit{P. Diaconis} and \textit{D. Freedman} [ibid. 8, 745-764 (1980; Zbl 0434.60034)] on the distance between a finite exchangeable sequence and a mixture of finite sequences of i.i.d. variables.
scientific article; zbMATH DE number 3984220