Limit theorems for branching Markov processes

We establish almost sure limit theorems for a branching symmetric Hunt process in terms of the principal eigenvalue and the ground state of an associated Schrödinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying Hunt process X. The almost sure limit theorems are established under the assumption that the associated Schrödinger operator of X has a spectral gap. Such an assumption is satisfied if the underlying process X is a Brownian motion, a symmetric α-stable-like process on or a relativistic symmetric stable process on .     

Limit theorems for flows of branching processes

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.     

Central limit theorems for supercritical branching Markov processes

In this paper we establish spatial central limit theorems for a large class of supercritical branching Markov processes with general spatial-dependent branching mechanisms. These are generalizations of the spatial central limit theorems proved in [1] for branching OU processes with binary branching mechanisms. Compared with the results of [1], our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate.     

L ∞ — Limit theorems for Markov processes

A necessary and sufficient condition for convergence of Markov processesL _∞ is given. As a consequence we get a theorem concerning the convergence of Harris processes. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

Limit theorems for the integrals of some branching processes

A number of limit theorems for the integral of a non-supercritical age-dependent branching process with immigration are found. Some results are given for the subcritical case without immigration, but conditioned to stay positive. Finally a central limit theorem is given for the population size of the subcritical immigration set up under a condition when no limiting distribution exists.     

Limit theorems for recurrent semi-Markov processes and Markov renewal processes

For Harris recurrent Markov renewal processes and semi-Markov processes one obtains a central limit theorem. One also obtains Berry-Esseen type estimates for this theorem. Their proof is based on the Kolmogorov-Doeblin regenerative method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 86–97, 1985.     

Limit theorems in critical Galton-Watson branching processes with migration

We study critical Galton-Watson branching processes with migration. It is assumed that the second moment of the number of direct descendants is infinite. Limit theorems are proved for the case where the mean value of migration is equal to zero. The work generalizes the results obtained by Nagaev and Khan [Teor. Veroyatn. Ee Primen.,25, No. 3, 523–534 (1980)] for the case where the second moment is finite.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1307–1317, October, 1995.     

Polling systems and multitype branching processes

The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.This work was done while the author was at the Centre for Mathematics and Computer Science (CWI) in Amsterdam, The Netherlands.     

Linear functionals for critical multitype galton-watson branching processes

We consider a critical K-type Galton-Watson branching process {Z(t)=(Z₁(t),…,Z_K(t)): t=0,1,…}. It is well known that, under rather general assumptions on the characteristics of the branching process, for any real vector the distribution of the sequence of sums , properly scaled and given thatZ(t)≠0 converges to a limit law as t→∞. In addition, the scaling function is of order t if the variances of the number of direct descendants of particles of all types are finite. But the limiting distribution has a unit atom at zero if the vectorw is orthogonal to the left eigenvector of the mean matrix of the process corresponding to its Perron root. If the variances of the number of direct descendants of particles of all types are finite, then to get a nontrivial limiting distribution for suchw (under the condition of nonextinction) one should always scaleZ(t)w by a function proportional to . In the case where the variances of the number of direct descendants of some types are infinite, the order of a scaling function providing existence of a nontrivial limit essentially depends onw. In the present note, we take the next step, namely, for a large class of processes with K≥3 types of particles and infinite variances of the number of direct descendants, we show that one can find two vectorsw ₁ andw ₂ orthogonal to the mentioned left eigenvector, such that the processesZ(t)w ₁ andZ(t)w ₂ conditioned on nonextinction up to moment t have different orders of growth in t as t→∞. Supported by the Russian Foundation for Basic Research (grant No. 96-15-96092). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of L^p-continuity for some 1 ? p ? 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.     

Critical multitype branching processes with infinite variance

The exponential limit law for the critical multitype Bienaymé-Galton-Watson process is extended to a class of offspring distributions some or all of whose second moments are infinite. Several asymptotic consequences pertaining to transition probabilities and invariant measures are derived.     

Convergence of critical multitype Galton-Watson branching processes

Allowing an offspring probability distribution that has infinite variances, we establish the convergence in finite-dimensional distributions of normalized critical multitype Galton-Watson branching processes with increasing initial population size in the two cases of not conditioning and of conditioning on non-extinction of the processes in the nth generation. Furthermore, if the offspring probability distribution has only finite variances, we show that some linear functions of the above processes weakly converge to the diffusions given by Feller, and by Lamperti and Ney.