RATE OF CONVERGENCE OF EMPIRICAL MEASURES FOR EXCHANGEABLE SEQUENCES

Let S be a finite set, (X-n) an exchangeable sequence of S-valued random variables, and mu(n) = (1/n) Sigma(n)(i=1) delta X-i the empirical measure. Then, mu(n) (B) ->(a.s.) mu (B) for all B subset of S and some (essentially unique) random probability measure mu. Denote by L(Z) the probability distribution of any random variable Z. Under some assumptions on L(mu), it is shown that (a)(n) <= rho[L(mu(n)), L(mu)] <=(b)(n) and rho[L(mu(n)), L(a(n))] <= (c)(nu) where rho is the bounded Lipschitz metric and a(n)(center dot) = P(Xn+1 is an element of center dot vertical bar X-1,..., X-n) is the predictive measure. The constants a, b,c > 0 and u is an element of (1/2, 1] depend on L(mu) and card(S) only. (C) 2017 Mathematical Institute Slovak Academy of Sciences