Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles
Abstract
Let {Xn:n≥0} be a controlled branching process with controls ϕn, and let Nn=ϕn(Xn) be its progenitor count. We study large deviations of Xn+1/Nn, Nn/Xn, and Xn+1/Xn, on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur--Rao--Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental--boundary--persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact n−1/2-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.
Related
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BibTeX
@misc{gonzalez2026large,
title = {{Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles}},
author = {Miguel González and Carmen Minuesa and Inés del Puerto and Anand N. Vidyashankar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23696},
eprint = {2609.23696},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23696},
}
RIS
TY - UNPB
TI - Large Deviations for Controlled Branching Processes: Random Slopes, Harmonic Moments, and Transfer Principles
AU - Miguel González
AU - Carmen Minuesa
AU - Inés del Puerto
AU - Anand N. Vidyashankar
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23696
UR - https://arxiv.org/abs/2609.23696
AB - Let {X_n:n≥0} be a controlled branching process with controls ϕ_n, and let N_n=ϕ_n(X_n) be its progenitor count. We study large deviations of X_n+1/N_n, N_n/X_n, and X_n+1/X_n, on their natural positive-denominator events, in supercritical and critical regimes. We treat generation-wide random slopes and asymptotically deterministic first-order slopes. For asymptotically affine random-slope controls, an affine recursion and a random environmental product yield a sharp Bahadur--Rao--Petrov upper-tail asymptotic. A positive Cramér tilt identifies the prefactor via a nontrivial normalized limit, and conditional on the upper-deviation event the three ratios concentrate around their environmental centers. With individual-sum controls, the process is an exact branching process in a random environment with immigration (BPREI). Motivated by this connection, we establish sharp upper large-deviation asymptotics for BPREI and transfer its positive-transform and harmonic-moment profiles to the controlled process. In the nondecreasing supercritical case, harmonic moments exhibit an exact environmental--boundary--persistence trichotomy. In a centered finite-variance critical class, finite-horizon-minimum methods yield exact n^−1/2-scale limits for all positive harmonic orders and for the three ratio-deviation probabilities. For deterministic first-order slopes, the environmental-product mechanism disappears. Matched lower and upper probability-generating-function envelopes transfer auxiliary exact-recursion profiles to controls without an exact branching recursion, producing distinct supercritical and critical harmonic-moment trichotomies. Thus the location of first-order randomness determines the lower-tail mechanism, while immigration may change a sharp constant, a phase boundary, or the decay exponent.
ER -