Minimax and adaptive inference for branching ratios under long memory
Abstract
The branching ratio ρ of a self-exciting point process determines the branching margin 1−ρ to the critical boundary. We study inference on ρ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order T−γ/(2γ+1) for a power-law envelope with tail exponent γ. A block-dispersion estimator whose block length grows as T1/(2γ+1) attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown γ, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.
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BibTeX
@misc{herreramarin2026minimax,
title = {{Minimax and adaptive inference for branching ratios under long memory}},
author = {Mauricio Herrera-Marín},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38267},
eprint = {2609.38267},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38267},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimax and adaptive inference for branching ratios under long memory
AU - Mauricio Herrera-Marín
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38267
UR - https://arxiv.org/abs/2609.38267
AB - The branching ratio ρ of a self-exciting point process determines the branching margin 1−ρ to the critical boundary. We study inference on ρ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order T^−γ/(2γ+1) for a power-law envelope with tail exponent γ. A block-dispersion estimator whose block length grows as T^1/(2γ+1) attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown γ, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.
ER -