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

Strong Law of Large Numbers for Branching Diffusions

Let X be the branching particle diffusion corresponding to the operator Lu + β(u2 − u) on D ⊆ Rd (where β ≥ 0 and β 6≡ 0). Let λc denote the generalized principal eigenvalue for the operator L + β on D and assume that it is finite. When λc > 0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology ...

Local Limit Theory and Large Deviations for Supercritical Branching Processes

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn :n ≥ 1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P (Zn = vn) as vn ր ∞, and use this to study conditional large deviations of {YZn :n ≥ 1}, where Yn satisfies an LDP, particularly of {Z −1 n Zn+1...

Immigration Superprocesses with Dependent Spatial Motion and Non-critical Branching

A Note on the Strong Law of Large Numbers

Petrov (1996) proved the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of pairwise independent and identically distributed random variables. This note examines this connection to a sequence of pairwise negative quadrant dependent (NQD) and identically distributed random variables. As a consequence of the main theorem ...

Optimal local Hölder index for density states of superprocesses with 1+-branching mechanism

Received May 2008; revised April 2009. Supported by the German Israeli Foundation for Scientific Research and Development, Grant G-807-227.6/2003. AMS 2000 subject classifications. Primary 60J80; secondary 60G57.