Discrete Time Branching Processes in Random Environment

Abstract

Branching processes (Z n ) n≥0 in varying environment generalize the Galton-Watson process, in that they allow time-dependence of the offspring distribution. Our main results concern general criteria for a.s. extinction, square-integrability of the martingale (Z n /E[Z n ]) n≥0 , properties of the martingale limit W and a Yaglom type result stating convergence to an exponential limit distribution of the suitably normalized population size Z n , conditioned on the event Z n > 0. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

“…Note that subcritical BPRE's may be additionally split into several other classes with essentially different properties (see [5] and [10] for more detail). In particular, a subcritical BPRE is called strongly subcritical if E Xe X < 0, intermediate subcritical if E Xe X = 0, and weakly subcritical if there is a number 0 < β < 1 such that…”

A revisited proof of the Seneta-Heyde norming for branching random walks under optimal assumptions

“…In order to prove Lemma 4.3, we introduce in Lemma 5.1 a result on convergence of functionals of the branching random walk conditioned on the spine staying above the origin until time n to corresponding functionals of the process conditioned on the spine staying above the origin for all time. It is inspired by a analogous result for random walks by Kersting and Vatutin [15,Lemma 5.2]. We recall that R is harmonic for the sub-Markov process obtained by killing (X ξn ) n≥0 when entering (−∞, 0) (Proposition 2.2).…”

“…Equations (2.4) and (2.5) in Lemma 2.3 are due to Kozlov [17]. Equation (2.6) can be found for example in Aïdékon and Shi [2] and is derived there as a consequence of Feller's renewal theorem and Sparre Andersen's identities for random walks (see for example [15], section 4.2). A different approach, in the spirit of Madaule [21], is to use an invariance principle for the Doob R-transform of the random walk killed below the origin, see Section 4 for a precise definition of this process.…”

“…• a lemma by Kersting and Vatutin [15] on the convergence of functionals of random walks conditioned to stay above the origin until a finite time n to analogous functionals of random walks conditioned to stay above the origin for all time.…”

The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

“…Using (14) we conclude that |e i L n,1 1| |L n,1 e T i | ≥ min q,r l n,1 (q, r) p max q,r l n,1 (q, r) ≥ 1 p∆ 2 .…”