No Arabic abstract
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 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 describe precise asymptotics of upper large deviations, i.e. $mathbb{P}[Z_n > e^{rho n}]$. Moreover in the subcritical case, under the Cramer condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.
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 $log Z_n$.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d ge 3$ and the environment is not too random, then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d le 2$, or the environment is ``random enough, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of replica overlap. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.
In this paper, we are concerned with SIR epidemics in a random environment on complete graphs, where every edges are assigned with i.i.d. weights. Our main results give large and moderate deviation principles of sample paths of this model.
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.