Abstract
We consider the stochastic behavior of a class of local U-statistics of Poisson processes—which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior—for point clouds lying in diverging halfspaces. We provide limit theorems for distributions with light and heavy tails. In particular, we prove finite-dimensional central limit theorems. In the light tail case we investigate tails that decay at least as slow as exponential and at least as fast as Gaussian. These results also furnish as a corollary that U-statistics for halfspaces diverging at different angles are asymptotically independent, and that there is no asymptotic independence for heavy-tailed densities. Using state-of-the-art bounds derived from recent breakthroughs combining Stein’s method and Malliavin calculus, we quantify the rate of this convergence in terms of Kolmogorov distance. We also investigate the behavior of local U-statistics of a Poisson process conditioned to lie in a diverging halfspace and find that the upper bound on the Kolmogorov distance to a standard normal distribution is smaller the lighter the tail of the density is.
Funding Statement
The author gratefully acknowledges partial financial support from the NSF: 1934985 and 1940124.
Acknowledgments
The author would like to thank the two anonymous referees for their valuable feedback, which greatly aided in the presentation of this article. The author would also like to thank Takashi Owada for his insightful feedback on an early draft of this paper.
Citation
Andrew M. Thomas. "Central limit theorems and asymptotic independence for local U-statistics on diverging halfspaces." Bernoulli 29 (4) 3280 - 3306, November 2023. https://doi.org/10.3150/23-BEJ1583
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