MOMENTS AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION IN RANDOM ENVIRONMENTS

Let(Z_(n))be a branching process with immigration in a random environmentξ,whereξis an independent and identically distributed sequence of random variables.We show asymptotic properties for all the moments of Z_(n) and describe the decay rates of the n-step transition probabilities.As applications,a large deviation principle for the sequence log Z_(n) is established,and related large deviations are also studied.

CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT

We are interested in the convergence rates of the submartingale Wn=Z_(n)/Π_(n)to its limit W,where(Π_(n))is the usually used norming sequence and(Z_(n))is a supercritical branching process with immigration(Y_(n))in a stationary and ergodic environmentξ.Under suitable conditions,we establish the following central limit theorems and results about the rates of convergence in probability or in law:(i)W-W_(n) with suitable normalization converges to the normal law N(0,1),and similar results also hold for W_(n+k)-W_(n) for each fixed k∈N^(*);(ii)for a branching process with immigration in a finite state random environment,if W_(1) has a finite exponential moment,then so does W,and the decay rate of P(|W-W_(n)|>ε)is supergeometric;(iii)there are normalizing constants an(ξ)(that we calculate explicitly)such that a_(n)(ξ)(W-W_(n))converges in law to a mixture of the Gaussian law.

Precise Asymptotics in Limit Theorems for a Supercritical Branching Process with Immigration in a Random Environment

Let(Zn)be a supercritical branching process with immigration in an independent and identically distributed random environment.Under necessary moment conditions,we show the exact convergence rate in the central limit theorem on log Zn and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm.By similar approach and with the help of a change of measure,we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations.

Limit theorems for a supercritical branching process with immigration in a random environment

Let(Z_n) be a supercritical branching process with immigration in a random environment. Firstly, we prove that under a simple log moment condition on the offspring and immigration distributions, the naturally normalized population size W_n converges almost surely to a finite random variable W. Secondly, we show criterions for the non-degeneracy and for the existence of moments of the limit random variable W. Finally, we establish a central limit theorem, a large deviation principle and a moderate deviation principle about log Z_n.

Almost sure, L1-and L2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

Under a first order moment condition on the immigration mechanism,we show that an appropriately scaled supercritical and irreducible multi-type continuous state and continuous time branching process with immigration(CBI process)converges almost surely.If an x log(x)moment condition on the branching mechanism does not hold,then the limit is zero.If this x log(x)moment condition holds,then we prove L1 convergence as well.The projection of the limit on any left non-Perron eigenvector of the branching mean matrix is vanishing.If,in addition,a suitable extra power moment condition on the branching mechanism holds,then we provide the correct scaling for the projection of a CBI process on certain left non-Perron eigenvectors of the branching mean matrix in order to have almost sure and L1 limit.Moreover,under a second order moment condition on the branching and immigration mechanisms,we prove L2 convergence of an appropriately scaled process and the above-mentioned projections as well.A representation of the limits is also provided under the same moment conditions.

Time to most recent common ancestor for stationary continuous state branching processes with immigration

Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables： the time to the most recent common ancestor （MRCA）, the size of the current population, and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established. These generalize the results obtained by Y. T. Chen and J. F. Delmas.

A Fluctuation Type Limit Theorem for Ji(?)ina Processes with Immigration

It is proved by the theory of semigroup that the Ornstein-Uhlenbeck type process with jumps can arise from the fluctuation limit of a sequence of Jirina processes with immigration under suitable moments conditions.