Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 1, 24--44

Finitely Additive Uniform Limit Theorems

Patrizia Berti, Universita' di Modena e Reggio-Emilia, Modena, Italy
Pietro Rigo, Universita' di Pavia,  Pavia, Italy

SUMMARY. Some finitely additive limit theorems, which do not need a strategic setting, are proved. Let $(X_n)$ be a sequence of random variables, $\mu_n=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$ and $a_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\dots,X_n)$, where all probability measures (both conditional and unconditional) are assessed according to de Finetti's coherence principle. In the main result, connected with Bayesian predictive inference, conditions for $\sup_{A\in\mathcal{D}}\abs{\mu_n(A)-a_n(A)}\rightarrow 0$ in probability are given, where $\mathcal{D}$ is any class of events. Under mild assumptions, it is also shown that $\sup_{A\in\mathcal{D}}\abs{\mu_n(A)-\mu_m(A)}\rightarrow 0$, in probability, whenever $(X_n)$ has stationary finite dimensional distributions. Further, asymptotic exchangeability of a certain class of sequences is proved, and this allows to extend a characterization of exchangeability due to Kallenberg (1988).

AMS (1991) subject classification. Primary 60A05, 60A10, 60F15.

Key words and phrases. Coherence, empirical measure, exchangeability,  finitely additive probability, predictive measure,  strategic probability, uniform limit theorem.

Full paper (PDF)