Local limit theory and large deviations for supercritical Branching processes (original) (raw)
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Harmonic moments and large deviation rates for supercritical branching processes
Annals of Applied Probability, 2003
Let {Z n , n ≥ 1} be a single type supercritical Galton-Watson process with mean EZ 1 ≡ m, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence {R n ≡ Z n+1 Z n : n ≥ 1} and establishes a "phase transition" in rates depending on whether r, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant α. This is done via a careful analysis of the harmonic moments of Z n .
Some asymptotic results for near critical branching processes
Near critical single type Bienaymé-Galton-Watson (BGW) processes are considered. It is shown that, under appropriate conditions, Yaglom distributions of suitably scaled BGW processes converge to that of the corresponding diffusion approximation. Convergences of stationary distributions for Q-processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established as well. Although most of the work is concerned with the single type case, similar results for multitype settings can be obtained. As an illustration, convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is established.
Stochastic Processes and their Applications, 1999
Let (Z(t): t¿0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)=EZ(t)(t → ∞) or that of Yn (n → ∞). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z6x) as x → 0, and that of the characteristic function Ee itZ and its derivative as |t| → ∞; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z ¿ x) as x → ∞, the analyticity of the characteristic function (t) = Ee itZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.
On the Existence of ?-Moments of the Limit of a Normalized Supercritical Galton?Watson Process
Journal of Theoretical Probability, 2004
Let (Z n) n 0 be a supercritical Galton-Watson process with finite reproduction 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 < Eφ(W) < ∞ if, and only if, ELφ(Z 1) < ∞ , where We further show that functions φ(x) = x α L(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 Lφ(x) grows at the same order of magnitude as φ(x) so that ELφ(Z 1) < ∞ and Eφ(Z 1) < ∞ are equivalent. However, α = 1 implies lim x→∞ Lφ(x)/φ(x) = ∞ and hence that ELφ(Z 1) < ∞ is a strictly stronger condition than Eφ(Z 1)<∞. If φ(x) = x log p x for some p >0 it can be shown that Lφ(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 EZ 1 log Z 1 < ∞ and EW > 0 are equivalent.
Laws of large numbers for supercritical branching Gaussian processes
Stochastic Processes and their Applications, 2018
A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein-Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized. Contents 1. Introduction 1 2. Model and main result 3 3. Moment formulae 8 4. The weak law of large numbers 12 5. The strong law of large numbers 14 6. Examples 22 7. Acknowledgements 36 References 36
Limit Theorems for Exceedances of Sequence of Branching Processes
The Bulletin of the Malaysian Mathematical Society Series 2
A problem of the first exceedance of a given level by the family of independent branching processes with and without immigration is considered. Using limit theorems for large deviations for processes with and without immigration limit theorems for the index of the first process exceeding some fixed or increasing levels in critical, subcritical and supercritical cases are proved. Asymptotic formulas for the expectation of the index are also obtained.
On laws of large numbers in L^2 for supercritical branching Markov processes beyond λ-positivity
2017
We give necessary and sufficient conditions for laws of large numbers to hold in L^2 for the empirical measure of a large class of branching Markov processes, including λ-positive systems but also some λ-transient ones, such as the branching Brownian motion with drift and absorption at 0. This is a significant improvement over previous results on this matter, which had only dealt so far with λ-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
We give necessary and sufficient conditions for laws of large numbers to hold in L 2 for the empirical measure of a large class of branching Markov processes, including λ-positive systems but also some λ-transient ones, such as the branching Brownian motion with drift and absorption at 0. This is a significant improvement over previous results on this matter, which had only dealt so far with λ-positive systems. Our approach is purely probabilistic and is based on spinal decompositions and many-to-few lemmas. In addition, we characterize when the limit in question is always strictly positive on the event of survival, and use this characterization to derive a simple method for simulating (quasi-)stationary distributions.