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Register a new userLimit theorems for supercritical branching processes in random environment
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We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
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The work continues the authors many-year research in theory of maximal branching processes, which are obtained from classical branching processes by replacing the summation of descendant numbers with taking the maximum. One can say that in each generation, descendants of only one particle survive, namely those of the particle that has the largest number of descendants. Earlier, the author generalized processes with integer values to processes with arbitrary nonnegative values, investigated their properties, and proved limit theorems. Then processes with several types of particles were introduced and studied. In the present paper we introduce the notion of maximal branching processes in random environment (with a single type of particles) and an important case of a power-law random environment. In the latter case, properties of maximal branching processes are studied and the ergodic theorem is proved. As applications, we consider gated infinite-server queues.
In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:nge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n earrow infty, and use this to study conditional large deviations of {Y_{Z_n}:nge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:nge1} conditioned on Z_nge v_n.
Let $left { Z_{n}, nge 0 right }$ be a supercritical branching process in an independent and identically distributed random environment $xi =left ( xi _{n} right )_{ngeq 0} $. In this paper, we get some deviation inequalities for $ln left (Z_{n+n_{0} } / Z_{n_{0} } right ).$ And some applications are given for constructing confidence intervals.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of lilypads built on a Poisson point process in $mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.
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