Abstract
Let $X^{(i)}_j, i = 1,\ldots, k; j \in \mathbf{N}$, be independent $d$-dimensional random vectors which are identically distributed for each fixed $i = 1,\ldots, k$. We give a sufficient condition for almost sure convergence of a sequence $T_{n_1,\ldots, n_k}$ of statistics based on $X^{(i)}_j i = 1,\ldots, k; j = 1, \ldots, n_i$, which are symmetric functions of $X^{(i)}_1,\ldots, X^{(i)}_{n_i}$ 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.
Citation
N. Henze. B. Voigt. "Almost Sure Convergence of Certain Slowly Changing Symmetric One- and Multi-Sample Statistics." Ann. Probab. 20 (2) 1086 - 1098, April, 1992. https://doi.org/10.1214/aop/1176989819
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