Conceptual proofs of L log L criteria for mean behavior of branching processes

@article{Lyons1995ConceptualPO,
  title={Conceptual proofs of L log L criteria for mean behavior of branching processes},
  author={Russell Lyons and Robin Pemantle and Yuval Peres},
  journal={Annals of Probability},
  year={1995},
  volume={23},
  pages={1125-1138}
}
The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations. We give conceptual proofs of these theorems based on comparisons of Galton-Watson measure to another measure on the space of trees. This approach also explains Yaglom's… 
View via Publisher
doi.org
416 Citations
Highly Influential Citations
40
Background Citations
123
Methods Citations
89
Results Citations
8

Figures from this paper

416 Citations

Citation Type

Citation Type

The xlogx condition for general branching processes
The x log x condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the…
Sharpness of Second Moment Criteria for Branching and Tree-Indexed Processes
A class of branching processes in varying environments is exhibited which become extinct almost surely even though the means Mn grow fast enough so that \( \sum {M_n^{ - 1}} \) is finite. In fact,…
New rates for exponential approximation and the theorems of R\'{e}nyi and Yaglom
We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain rates of…
L Log L Criterion for a Class of Superdiffusions
In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching…
Condition for Supercritical Branching Hunt Processes
In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum L logL theorem for branching Hunt processes. This result is a generalization of the results…
Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks
We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of…
New rates for exponential approximation and the theorems of Rényi and Yaglom
We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of…
Llog L Condition for Supercritical Branching Hunt Processes
In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum Llog L theorem for branching Hunt processes. This result is a generalization of the results…
Elementary new proofs of classical limit theorems for Galton-Watson processes
Classical results describe the asymptotic behaviour of a Galton-Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The…
Galton–Watson Trees
We recall a few elementary properties of supercritical Galton–Watson trees, and introduce the notion of size-biased trees. As an application, we give in Sect. 2.3 the beautiful conceptual proof by…
...
...

References

SHOWING 1-10 OF 47 REFERENCES
Growing conditioned trees
The occurrence of large values in stationary sequences
SummaryA direct proof is given of the Tanny (1974) result that for certain non-decreasing sequences an, it is true that lim supan−1Xn = 0 or + ∞ with probability one for all ergodic stationary…
A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means
LENGTH‐BIASING, CHARACTERIZATIONS OF LAWS AND THE MOMENT PROBLEM
Summary A positive random variable X with a finite mean has an induced length-biased law represented by Y, and Y is stochastically larger than X. An independent uniform random contraction of Y, UY,…
A CASCADE DECOMPOSITION THEORY WITH APPLICATIONS TO MARKOV AND EXCHANGEABLE CASCADES
A multiplicative random cascade refers to a positive T -martingale in the sense of Kahane on the ultrametric space T = {0, 1, . . . , b− 1}. A new approach to the study of multiplicative cascades is…
Branching processes with random environments
We postulate that {Zn} satisfy the recurrence relations (5) E(s^\Fn,z) = K W ] ^ a.s. and for any set of integers l ^ W i < w 2 < • • • <rik with | s t | â l , A MS Subject Classifications. Primary…
Probability: Theory and Examples
This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a…
KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees
SummaryIf Rt is the position of the rightmost particle at time t in a one dimensional branching brownian motion, u(t, x)=P(Rt>x) is a solution of KPP equation: $$\frac{{\partial u}}{{\partial t}} =…
Subdiffusive behavior of random walk on a random cluster
On considere deux cas de marche aleatoire {X n } n≥0 sur un graphe aleatoire #7B-G. Dans le cas ou #7B-G est l'arbre d'un processus de branchement critique, conditionne par la non-extinction, si h(x)…
Trees Generated by a Simple Branching Process
...
...