Supercritical Branching Random Walk Research Articles

A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

Abstract Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_∞(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W∞(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W∞(θ)-Wn(θ) as n→∞.

A functional central limit theorem for branching random walks, almost sure weak convergence and applications to random trees

Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_{n}(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the process \[D_{n}(u):=m^{\frac{1}{2}n}(W_{\infty}(\frac{u}{\sqrt{n}})-W_{n}(\frac{u}{\sqrt{n}}))\] converges weakly, on a suitable space of analytic functions, to a Gaussian random analytic function with random variance. Using this result, we prove central limit theorems for the total path length of random trees. In the setting of binary search trees, we recover a recent result of R. Neininger [Random Structures Algorithms 46 (2015) 346–361], but we also prove a similar theorem for uniform random recursive trees. Moreover, we replace weak convergence in Neininger’s theorem by the almost sure weak (a.s.w.) convergence of probability transition kernels. In the case of binary search trees, our result states that \[\mathcal{L}\{\sqrt{\frac{n}{2\log n}}(\operatorname{EPL}_{\infty}-\frac{\operatorname{EPL}_{n}-2n\log n}{n})\Big |\mathcal{G}_{n}\}\overset{\mathrm{a.s.w.}}{\underset{n\to\infty}\longrightarrow}\{\omega \mapsto\mathcal{N}_{0,1}\},\] where $\operatorname{EPL}_{n}$ is the external path length of a binary search tree $X_{n}$ with $n$ vertices, $\operatorname{EPL}_{\infty}$ is the limit of the Régnier martingale and $\mathcal{L}\{\cdot |\mathcal{G}_{n}\}$ denotes the conditional distribution w.r.t. the $\sigma$-algebra $\mathcal{G}_{n}$ generated by $X_{1},\ldots,X_{n}$. Almost sure weak convergence is stronger than weak and even stable convergence. We prove several basic properties of the a.s.w. convergence and study a number of further examples in which the a.s.w. convergence appears naturally. These include the classical central limit theorem for Galton–Watson processes and the Pólya urn.

Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $\lambda^{*}n^{1/3}$ for some explicit constant $\lambda^{*}$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.

On the maximal displacement of subcritical branching random walks

We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each $$n\in \mathbb {N},$$ let $$M_{n}$$ be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists $$\rho >1$$ such that the function $$\begin{aligned} g(c,n):=\rho ^{cn} P(M_{n}\ge cn), \quad \hbox {for each }c>0 \hbox { and } n\in \mathbb {N}, \end{aligned}$$ satisfies the following properties: there exist $$0<\underline{\delta }\le \overline{\delta } < {\infty }$$ such that if $$c<\underline{\delta }$$ , then $$\begin{aligned} 0<\liminf _{n\rightarrow \infty } g (c,n)\le \limsup _{n\rightarrow \infty } g (c,n) {\le 1}, \end{aligned}$$ while if $$c>\overline{\delta }$$ , then $$\begin{aligned} \lim _{n\rightarrow \infty } g (c,n)=0. \end{aligned}$$ Moreover, if the jump distribution has a finite right range R, then $$\overline{\delta } < R$$ . If furthermore the jump distribution is “nearly right-continuous”, then there exists $$\kappa \in (0,1]$$ such that $$\lim _{n\rightarrow \infty }g(c,n)=\kappa $$ for all $$c<\underline{\delta }$$ . We also show that the tail distribution of $$M:=\sup _{n\ge 0}M_{n}$$ , namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at $$\underline{\delta }$$ ). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.

A weakly supercritical mode in a branching random walk

The case of weakly supercritical branching random walks is considered. A theorem on asymptotic behavior of the eigenvalue of the operator defining the process is obtained for this case. Analogues of the theorems on asymptotic behavior of the Green function under large deviations of a branching random walk and asymptotic behavior of the spread front of population of particles are established for the case of a simple symmetric branching random walk over a many-dimensional lattice. The constants for these theorems are exactly determined in terms of parameters of walking and branching.

The structure of the positive discrete spectrum of the evolution operator arising in branching random walks

A branching random walk (BRW) with continuous time and a finite number of branching sources located at points of a multidimensional lattice is considered. The definition of weakly supercritical BRWs, whose discrete spectrum contains a unique positive eigenvalue, is introduced. Conditions for a supercritical BRW to be weakly supercritical are determined.

The Seneta–Heyde scaling for the branching random walk

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical branching random walk, and study the additive martingale $(W_n)$. We prove that, upon the system's survival, $n^{1/2}W_n$ converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544--581], of the derivative martingale.

Convergence in law of the minimum of a branching random walk

We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044-1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.

Limiting laws of supercritical branching random walks

In this Note, we make explicit the limit law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in Barral et al. (2012) [5] for a continuous analogue of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectures formulated by physicists (Derrida and Spohn, 1988 [9]) about the Poisson–Dirichlet nature of the jumps in the limit, and precising the conjecture by giving the spatial distribution of these jumps.

Supercritical Branching Random Walks with a Single Source

We consider two continuous-time branching random walks on multidimensional lattices with birth and death of particles at the origin. In the first one, the underlying random walk is assumed to be symmetric. In the second one, an additional parameter is introduced to intensify artificially the prevalence of branching or walk at the origin. As a side effect, it violates symmetry of the random walk. Necessary and sufficient conditions for exponential growth for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained. General methods to study the models in the supercritical case are proposed.

Weak convergence for the minimal position in a branching random walk: A simple proof

Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after n steps behaves in probability like $$ \frac{3} {2} $$ log n when n → ∞. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables.

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 Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

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 Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609–631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817–840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks

Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.

Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

Let \(\mathcal{M}^{(n)} \), n = 1, 2, ..., be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to \(\mathcal{M}^{(n)} \) converges almost surely and in the mean to a random variable W. For a large subclass of nonnegative and concave functions ƒ, we provide a criterion for the finiteness of \(\mathbb{E}\) Wf(W). The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the ƒ-moments of perpetuities.

On the rate of convergence of a regular martingale related to a branching random walk

Let M n , n = 1, 2, ..., be a supercritical branching random walk in which the number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale W n related to M n is regular and W is a limit random variable. Let a(x) be a nonnegative function regularly varying at infinity with index greater than −1. We present sufficient conditions for the almost-sure convergence of the series $$\sum\nolimits_{n = 1}^\infty {a(n)(W - W_n )} $$ . We also establish criteria for the finiteness of EW ln+ Wa(ln+ W) and E ln+|Z ∞|a(ln+|Z ∞|), where $$Z_\infty : = Q_1 + \sum\nolimits_{n = 2}^\infty {M_1 \ldots M_n Q_{n + 1} } $$ and (M n , Q n ) are independent identically distributed random vectors not necessarily related to M n .

A prediction problem of the branching random walk

The location of the ‘favourite’ point at time T (T = 1, 2,…) of a supercritical branching random walk at ℤd is investigated.

A prediction problem of the branching random walk

The location of the ‘favourite’ point at time T (T = 1, 2,…) of a supercritical branching random walk at ℤd is investigated.