Kesten Stigum Research Articles

On the purity of the free boundary condition Potts measure on random trees

We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q -ary channels. The bounds for the corresponding threshold value of the inverse temperature are optimal for the Ising model and differ from the Kesten Stigum bound by only 1.50% in the case q = 3 and 3.65% for q = 4 , independently of d . Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.

A conceptual approach to a path result for branching Brownian motion

We give a new, intuitive and relatively straightforward proof of a path large-deviations result for branching Brownian motion (BBM) that can be thought of as extending Schilder’s theorem for a single Brownian motion. Our conceptual approach provides an elegant and striking new application of a change of measure technique that induces a ‘spine’ decomposition and builds on the new foundations for the use of spines in branching diffusions recently developed in Hardy and Harris [Robert Hardy, Simon C. Harris, A new formulation of the spine approach to branching diffusions, 2004, no. 0404, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-foundations.pdf; Robert Hardy, Simon C. Harris, Spine proofs for L p -convergence of branching-diffusion martingales, 2004, no. 0405, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-Lp-cgce.pdf], itself inspired by related works of Kyprianou [Andreas Kyprianou, Travelling wave solutions to the K–P–P equation: alternatives to Simon Harris’s probabilistic analysis, Ann. Inst. H. Poincaré Probab. Statist. 40 (1) (2004) 53–72] and Lyons et al. [Russell Lyons, A simple path to Biggins’ martingale convergence for branching random walk, in: Classical and Modern Branching Processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 217–221; Thomas Kurtz, Russell Lyons, Robin Pemantle, Yuval Peres, A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes, in: Classical and Modern Branching Processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 181–185; Russell Lyons, Robin Pemantle, Yuval Peres, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23 (3) (1995) 1125–1138]. Some of the techniques developed here will also apply in more general branching Markov processes, for example, see Hardy and Harris [Robert Hardy, Simon C. Harris, A spine proof of a lower bound for a typed branching diffusion, 2004, no. 0408, Mathematics Preprint, University of Bath. http://www.bath.ac.uk/~massch/Research/Papers/spine-oubbm.pdf].

On the Existence of ?-Moments of the Limit of a Normalized Supercritical Galton?Watson Process

Let (Z n ) n≥ 0 be a supercritical Galton–Watson process with finite re-production mean μ and normalized limit W=lim n → ∞μ−n Z n . Let further φ: [0,∞) → [0,∞) be a convex differentiable function with φ(0)=φ′(0)=0 and such that φ( $$x^{1/2^n}$$ ) is convex with concave derivative for some n ≥ 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E φ (W) < ∞ if, and only if, $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ , where $${\mathbb{L}}\phi (x){\mathop = \limits^{def}} \int_0^x {\int_0^s {\tfrac{{\phi '(r)}}{r}drds,\;\;\;x \geqslant 0.} }$$ We further show that functions φ(x)=x αL(x) which are regularly varying of order α ≤ 1 at ∞ are covered by this result if α ∉ {2 n : n ≥ 0 } and under an additional condition also if α=2 n for some n≥0. This was obtained in a slightly weaker form and analytically by Bingham and Doney. If α > 1, then $${\mathbb{L}}\phi (x)$$ grows at the same order of magnitude as φ(x) so that $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ and E φ(Z 1)< ∞ are equivalent. However, α=1 implies $$\lim _{x \to \infty } {\mathbb{L}}\phi (x)/\phi (x) = \infty$$ and hence that $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ is a strictly stronger condition than E φ(Z 1) < ∞. If φ(x)=x log p x for some p > 0 it can be shown that $${\mathbb{L}}\phi (x)$$ grows like x log p+1 x, as x→∞. For this special case the result is due to Athreya. As a by-product we also provide a new proof of the Kesten–Stigum result that E Z 1 log Z 1 0 are equivalent.

A strong law and a central limit theorem for controlled Galton-Watson processes

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.

A strong law and a central limit theorem for controlled Galton-Watson processes

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.

A limit theorem for population-size-dependent branching processes

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in therth mean and almost surely, are obtained for population-size-dependent branching processes.

A limit theorem for population-size-dependent branching processes

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.