**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.