Small positive values for supercritical branching processes in random environment

Vincent Bansaye; Christian Böinghoff

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 770-805
  • ISSN: 0246-0203

Abstract

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Branching Processes in Random Environment (BPREs) ( Z n : n ≥ 0 ) are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of ℙ ( 1 ≤ Z n ≤ k | Z 0 = i ) , k , i ∈ ℕ as n → ∞ . More precisely, we characterize the exponential decrease of ℙ ( Z n = k | Z 0 = i ) using a spine representation due to Geiger. We then provide some bounds for this rate of decrease. If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.

How to cite

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Bansaye, Vincent, and Böinghoff, Christian. "Small positive values for supercritical branching processes in random environment." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 770-805. <http://eudml.org/doc/272001>.

@article{Bansaye2014,
abstract = {Branching Processes in Random Environment (BPREs) $(Z_\{n\}\colon \ n\ge 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb \{P\}(1\le Z_\{n\}\le k\vert Z_\{0\}=i)$, $k,i\in \mathbb \{N\}$ as $n\rightarrow \infty $. More precisely, we characterize the exponential decrease of $\mathbb \{P\}(Z_\{n\}=k\vert Z_\{0\}=i)$ using a spine representation due to Geiger. We then provide some bounds for this rate of decrease. If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.},
author = {Bansaye, Vincent, Böinghoff, Christian},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {supercritical branching processes; random environment; large deviations; phase transitions},
language = {eng},
number = {3},
pages = {770-805},
publisher = {Gauthier-Villars},
title = {Small positive values for supercritical branching processes in random environment},
url = {http://eudml.org/doc/272001},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Bansaye, Vincent
AU - Böinghoff, Christian
TI - Small positive values for supercritical branching processes in random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 770
EP - 805
AB - Branching Processes in Random Environment (BPREs) $(Z_{n}\colon \ n\ge 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb {P}(1\le Z_{n}\le k\vert Z_{0}=i)$, $k,i\in \mathbb {N}$ as $n\rightarrow \infty $. More precisely, we characterize the exponential decrease of $\mathbb {P}(Z_{n}=k\vert Z_{0}=i)$ using a spine representation due to Geiger. We then provide some bounds for this rate of decrease. If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.
LA - eng
KW - supercritical branching processes; random environment; large deviations; phase transitions
UR - http://eudml.org/doc/272001
ER -

References

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