Functional limit theorems for weighted occupation times of supercritical superprocesses
Abstract
Suppose X=(Xt)t≥0 is a supercritical superprocess with a non-local branching mechanism on a Lusin space E, whose mean semigroup admits a positive leading eigenvalue λ1 with a bounded positive eigenfunction φ and a finite eigenmeasure φ̃. Let Yt:=∫0t Xsds be its occupation measure. We first prove a law of large numbers, that is, for every bounded measurable function f, e−λ1 t⟨ f,Yt⟩ converges in L2 to λ1−1⟨ f,φ̃⟩ Wφ∞, where Wφ∞ is the limit of the martingale Wφt:=e−λ1 t⟨φ,Xt⟩. We then establish functional central limit theorems for the fluctuations of ⟨ f,Yt⟩ around this random centering. We show that the normalisation and the limit are determined by the spectral decay rate of f compared with the critical value λ1/2, and there are three possible regimes: after suitable normalisation, the fluctuation process converges in distribution in 𝒞[0,+∞) either to a mixed Gaussian process √(Wφ∞)G(f), or to a constant Gaussian process with random variance depending on Wφ∞, or to a non-Gaussian limit given by the limit of a martingale associated with the spectral projection of f. These results are the occupation-time counterparts of the fluctuation theorems for instantaneous linear functionals of supercritical non-local branching superprocesses.
Related
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BibTeX
@misc{yang2026functional,
title = {{Functional limit theorems for weighted occupation times of supercritical superprocesses}},
author = {Ting Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12503},
eprint = {2610.12503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12503},
}
RIS
TY - UNPB
TI - Functional limit theorems for weighted occupation times of supercritical superprocesses
AU - Ting Yang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.12503
UR - https://arxiv.org/abs/2610.12503
AB - Suppose X=(X_t)_t≥0 is a supercritical superprocess with a non-local branching mechanism on a Lusin space E, whose mean semigroup admits a positive leading eigenvalue λ_1 with a bounded positive eigenfunction φ and a finite eigenmeasure φ̃. Let Y_t:=∫_0^t X_sds be its occupation measure. We first prove a law of large numbers, that is, for every bounded measurable function f, e^−λ_1 t⟨ f,Y_t⟩ converges in L^2 to λ_1^−1⟨ f,φ̃⟩ W^φ_∞, where W^φ_∞ is the limit of the martingale W^φ_t:=e^−λ_1 t⟨φ,X_t⟩. We then establish functional central limit theorems for the fluctuations of ⟨ f,Y_t⟩ around this random centering. We show that the normalisation and the limit are determined by the spectral decay rate of f compared with the critical value λ_1/2, and there are three possible regimes: after suitable normalisation, the fluctuation process converges in distribution in 𝒞[0,+∞) either to a mixed Gaussian process √(W^φ_∞)G(f), or to a constant Gaussian process with random variance depending on W^φ_∞, or to a non-Gaussian limit given by the limit of a martingale associated with the spectral projection of f. These results are the occupation-time counterparts of the fluctuation theorems for instantaneous linear functionals of supercritical non-local branching superprocesses.
ER -