Moderate Deviations for Random-Indexed Cluster Counts in Hierarchical Pitman-Yor Models
Abstract
Consider a sample of size N from a two-layer hierarchical Pitman--Yor model, and let ξ(N) denote the random number of clusters generated at the first layer of the hierarchy. We study moderate deviation principles for partition statistics evaluated at the random sample size ξ(N). In particular, we establish moderate deviation principles for the total number of clusters Kξ(N) and for the frequency count Ml,ξ(N), the number of upper-level clusters represented by exactly l first-level clusters. Our analysis combines fixed-sample-size small-tilt asymptotics with a moderate deviation principle for the random index ξ(N). A uniform small-tilt argument, together with exponentially weighted tail estimates, allows us to pass from deterministic sample sizes to the hierarchical random-index setting. We obtain explicit deviation speeds and good rate functions on a family of intermediate scales between the typical growth order Nα1α2 and the linear scale N. The resulting rate functions exhibit a common power-law structure governed by the product α1α2, revealing a multiplicative effect of the two levels of the hierarchy.
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BibTeX
@misc{yao2026moderate,
title = {{Moderate Deviations for Random-Indexed Cluster Counts in Hierarchical Pitman-Yor Models}},
author = {Nian Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05328},
eprint = {2610.05328},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05328},
}
RIS
TY - UNPB
TI - Moderate Deviations for Random-Indexed Cluster Counts in Hierarchical Pitman-Yor Models
AU - Nian Yao
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05328
UR - https://arxiv.org/abs/2610.05328
AB - Consider a sample of size N from a two-layer hierarchical Pitman--Yor model, and let ξ(N) denote the random number of clusters generated at the first layer of the hierarchy. We study moderate deviation principles for partition statistics evaluated at the random sample size ξ(N). In particular, we establish moderate deviation principles for the total number of clusters K_ξ(N) and for the frequency count M_l,ξ(N), the number of upper-level clusters represented by exactly l first-level clusters. Our analysis combines fixed-sample-size small-tilt asymptotics with a moderate deviation principle for the random index ξ(N). A uniform small-tilt argument, together with exponentially weighted tail estimates, allows us to pass from deterministic sample sizes to the hierarchical random-index setting. We obtain explicit deviation speeds and good rate functions on a family of intermediate scales between the typical growth order N^α_1α_2 and the linear scale N. The resulting rate functions exhibit a common power-law structure governed by the product α_1α_2, revealing a multiplicative effect of the two levels of the hierarchy.
ER -