Almost sure convergence of certain slowly changing symmetric one- and multi-sample statistics

Let X$_j^{i}$,i=1,…,k; j∈N, be independent d-dimensional random vectors which are identically distributed for each fixed i=1,…,k. We give a sufficient condition for almost sure convergence of a sequence T$_{n1,…,nk}$ of statistics based on X$_j^{i}$i=1,…,k;j=1,…,n$_i$, which are symmetric functions of X$_1^{i}$,…,X$_{ni}^{i}$ for each i and do not change too much when variables are added or deleted. A key auxiliary tool for proofs is the Efron-Stein inequality. Applications include strong limits for certain nearest neighbor graph statistics, runs and empty blocks.