Abstract: | Consider a family of square-integrable Rd-valued statistics Sk = Sk(X1,k1; X2,k2;…; Xm,km), where the independent samples Xi,kj respectively have ki i.i.d. components valued in some separable metric space Xi. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {Sk}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that Sk is symmetric in the coordinates of each sample Xi,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented. |