Rate of convergence of empirical measures for exchangeable sequences

Abstract

Abstract Let S be a finite set, (Xn ) an exchangeable sequence of S-valued random variables, and μn = (1/n) ∑ i = 1 n $\sum\limits_{i=1}^{n} $ δ Xi the empirical measure. Then, μn (B) ⟶ a . s . $\overset{a.s.}{\longrightarrow} $ μ(B) for all B ⊂ S and some (essentially unique) random probability measure μ. Denote by 𝓛(Z) the probability distribution of any random variable Z. Under some assumptions on 𝓛(μ), it is shown that a n ≤ ρ [ L ( μ n ) , L ( μ ) ] ≤ b n and ρ [ L ( μ n ) , L ( a n ) ] ≤ c n u $$\begin{array}{} \displaystyle \frac{a}{n}\le\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(\mu)\bigr]\le\frac{b}{n}\qquad\text{and}\qquad\rho\bigl[\mathcal{L}(\mu_n), \mathcal{L}(a_n)\bigr]\le\frac{c}{n^u} \end{array} $$ where ρ is the bounded Lipschitz metric and an (⋅) = P(X n+1 ∈ ⋅ | X 1, …, Xn ) is the predictive measure. The constants a, b, c > 0 and u ∈ ( 1 2 $\frac{1}{2} $ , 1] depend on 𝓛(μ) and card (S) only.