Refined Gibbs conditioning principle for certain infinite dimensional statistics

Let X-1, X-2, X-3,. . . be independent, identically distributed random obse rvations taking values in a Polish space Sigma, and theta a statistic on Si gma with values in a separable Banach space E. We examine the limit, law of (X-1, . . . , X-k) conditional on n(-1) Sigma(i=1)(n) theta(X-i) being in an open convex subset D of E. In this setting the conditional limit law is a k-fold product probability (P*)(k), where. P* is determined by the Gibbs conditioning principle. Our results describe the allowed dependence of k = k(n) on n in terms of explicit geometric conditions related to smoothness o f do at a dominating point.