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.     

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 multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

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.     

Law of large numbers for supercritical superprocesses with non-local branching

In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.     

Recurrence and transience of multitype branching random walks

We study a discrete time Markov process with particles being able to perform discrete time random walks and create new particles, known as branching random walk (BRW). We suppose that there are particles of different types, and the transition probabilities, as well as offspring distribution, depend on the type and the position of the particle. Criteria of (strong) recurrence and transience are presented, and some applications (spatially homogeneous case, Lamperti BRW, many-dimensional BRW) are studied.     

Persistence of critical multitype particle and measure branching processes

Summary We consider a class of systems of particles ofk types inR ^d undergoing spatial diffusion and critical multitype branching, where the diffusions, the particle lifetimes and the branching laws depend on the types. We prove persistence criteria for such systems and for their corresponding high density limits known as multitype Dawson-Watanabe processes. The main tool is a representation of the Palm distributions for a general class of inhomogeneous critical branching particle systems, constructed by means of a backward tree.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence) at late times for critical multitype spatially homogeneous branching particle systems in continuous time. Our conditions on the branching mechanism are close to the ones known from “classical” processes without motion component. This generalizes and complements results of López-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree analysis combined with the study of the long-term behavior of L ¹-norms of solutions of related systems of reaction-“diffusion” equations, which is perhaps also of some independent interest. Received: 13 August 1997 / Revised version: 12 May 1998 / Published online: 14 February 2000     

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;{\mathbf{P}}_{\mu }\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index ${\gamma }_0>1$. We first show that, under some conditions, ${\mathbf{P}}_{\mu }(\left|X_t\right|\ne 0)$ converges to 0 as $t\to \infty$ and is regularly varying with index ${({\gamma }_0-1)}^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t\left(f\right);{\mathbf{P}}_{\mu }\left(\cdot \right|6X_{t}6\ne 0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E\left[e^{-u{\mathbf{z}}^{({\gamma }_0-1)}}\right]=1-{(1+u^{-({\gamma }_0-1)})}^{-1∕({\gamma }_0-1)}.$     

Cut-by-curves criterion for the log extendability of overconvergent isocrystals

In this paper, we prove a ??cut-by-curves criterion?? for an overconvergent isocrystal on a smooth variety over a field of characteristic p?>?0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor, under certain assumption. This is a p-adic analogue of a version of cut-by-curves criterion for regular singularity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of ?-modules.     

On Koch's convergence criterion for branching continued fractions

Extension of the necessary condition for convergence of branching continued fractions with real positive elements to the case when the remainders are complex numbers leads to a multidimensional analog of Koch's theorem. However the part of the theorem that is known in the literature as the Stern-Stolz convergence criterion does not hold in this case.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 10–13.     

Spectral properties for a class of continuous state branching processes with immigration

We explicitly construct nontrivial invariant probability measures for a class of continuous state branching processes with immigration. The class of these measures include random Gamma measures and path space measures of Lévy subordinators as particular examples. Using the explicit construction we study long-time behaviour and hypercontractivity of the transition semigroups in corresponding L²-spaces.