2011/12/22 by Vincent Bansaye, Bansaye, Vincent, Christian Boeinghoff +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F17 #60J05 #60J80 #60K37 #92D25 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1112.5257
openalex publication_date 2011/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Branching Processes in Random Environment (BPREs) (Zn: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 ≤ Zn ≤ k|Z0=i), k,i∈ℕ as n→ ∞. More precisely, we characterize the exponential decrease of ℙ(Zn=k|Z0=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.