We consider critical reduced processes generated by critical Galton-Watson branching processes {Z(k), k≥ 0} in the case where the number offspring of one particle possibly has infinite variance. We obtain conditional limit theorems for such processes under the condition that {0

The limit theorems for subcritical and critical reduced processes generated by GaltonWatson branching processes starting with a large number of particles are obtained.

This paper considers a critical reduced process generated by a Galton-Watson branching process in which the number offspring of one particle possibly has infinite variance and conditional limit theorem proved. The rate of weak convergence of the critical reduced processes to the limit law is also obtained in the case when the number offspring of one particle has a finite variance.

We consider branching random processes with immigration starting from a random number of items. In this work, we provide estimates from above for the moments of such processes.

In the present paper, we give the upper bounds for moments and central moments of branching processes in a varying environment starting with a random number of particles.

We consider the critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of initial state on the asymptotic state of the process. For processes starting from large numbers of particles and satisfying condition (S), we prove the limit theorem similar to the result obtained by W. Feller. We also prove the theorem under the condition W(n) > 0 for the critical processes satisfying the conditions (S) and (M).

In this paper we consider critical Galton-Watson branching processes Zk, k ≥ 0 in the case when the number of direct offspring of one particle has infinite variance. Limit theorems for conditional distributions of Zk are proved.

In this paper, we consider a nearly critical branching process with immigration. We obtain the rate of convergence in central limit theorem for nearly critical branching processes with immigration.

We consider a sequence of branching processes with immigration, where the offspring mean tends to 1. Rates of growth and the asymptotic of fluctuation of common number individuals are investigated.

In the present paper we investigate a sequence of nearly critical branching process with immigration in the case when the mean of off springs number tends to 1 with the rate slowly than n(-1). We give conditions under which considered processes converge in probability to a determined process, and also prove limit theorems for the fluctuation of such processes.

We obtain the convergence rate for the distribution of the fluctuation of almost critical branching processes with immigration to the normal law.

We study a subcritical branching process with inhomogeneous immigration in the case where the mean value and variance of immigration are regularly varying at infinity. We show that a properly normalized subcritical process with immigration weakly approaches a deterministic process and prove the limit theorem for the fluctuation of this process.

Limit theorems are obtained for branching processes depending on the size of the population. These results contain the case of convergence of such processes to the deterministic limit. Functional central limit theorems for fluctuations are proved.

We study almost critical branching processes with infinitely increasing immigration and prove functional limit theorems for these processes.

A sequence of almost critical Galton–Watson branching processes with immigration is studied. Sufficient conditions for the weak convergence of such processes to a diffusion process are found.