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.