Scenery reconstruction with branching random walk

We study the problem of scenery reconstruction in arbitrary dimension using observations registered in boxes of size k (for k fixed), seen along a branching random walk. We prove that, using a large enough k for almost all the realizations of the branching random walk, almost all sceneries can be reconstructed up to equivalence.     

Minimal displacement of branching random walk

Summary Let denote a branching random walk in with mean particle productionm, m>1, and with incremental spatial distributionG, withG({0}) =p andG({1})=1–p. Ifmp=1, then the minimal displacement of behaves asymptotically like log logn/log 2. If the conditionG({1})=1–p is replaced byG((0, ))=1–p, we obtain a similar result.Research was partially supported by the National Science Foundation under grant MCS-7607039     

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 . We also establish criteria for the finiteness of EW ln⁺ Wa(ln⁺ W) and E ln⁺|Z _∞|a(ln⁺|Z _∞|), where and (M _n , Q _n ) are independent identically distributed random vectors not necessarily related to M _n . __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006.     

Limiting point processes in the branching random walk

Summary Take the nth generation of a supercritical branching random walk (a spatially homogeneous branching process) as a process of cluster centres and take independent copies of some simple point process Y as the clusters. Let the resulting point process be Y _n . For a given sequence of real numbers {x _n } let Y _n be centred on x _n . Under certain conditions, when an appropriate scale change is made, the resulting point process converges in distribution to a non-trivial limit.     

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.     

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n??th generation, R _n , has zero speed on the linear scale: R _n /n ?? 0 as n ?? ??. We further remove (??kill??) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Müller in Markov Process. Relat. Fields 12:805?C814, 2006; Hu and Shi in Ann. Probab. 37(2):742?C789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388?C412, 1998; and Power laws and killed branching random walks) that E [Z]?<??? while ${{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}$ . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.     

Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.     

Critical survival barrier for branching random walk

We consider a branching random walk with an absorbing barrier, where the associated one-dimensional random walk is in the domain of attraction of an α-stable law. We shall prove that there is a barrier and a critical value such that the process dies under the critical barrier, and survives above it. This generalizes previous result in the case that the associated random walk has finite variance.     

The survival of branching annihilating random walk

Summary Branching annihilating random walk is an interacting particle system on . As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown here that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non-attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by Gray and Griffeath is used.     

On the support of the simple branching random walk

Connectivity of the support of the simple branching random walk is established in certain asymmetric cases, extending a previous result of Grill.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

A note on exact convergence rate in the local limit theorem for a lattice branching random walk

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton-Watson process and the moving law of particles by a discrete random variable on the integer lattice Z. Denote by Z_n(z) the number of particles in the n-th generation in the model for each z ∈ Z. We derive the exact convergence rate in the local limit theorem for Z_n(z) assuming a condition like "EN(log N)~(1+λ) ∞" for the offspring distribution and a finite moment condition on the motion law. This complements the known results for the strongly non-lattice branching random walk on the real line and for the simple symmetric branching random walk on the integer lattice.     

Branching random walk with a critical branching part

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timek _n withk _n /n→β. We attach to each vertexv of this tree a random variableX(v) and define , where π(0,v) is the unique path in the family tree from its root tov. FinallyM _n is the maximal displacement of the branching random walkS(·), that isM _n =max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn ^−1/2 M _n converges in distribution. In particular {n ^−1/2 M _n } _n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.     

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.     

Spectral properties of evolutionary operators in branching random walk models

We introduce a model of continuous-time branching random walk on a finite-dimensional integer lattice with finitely many branching sources of three types and study the spectral properties of the operator describing the evolution of the mean numbers of particles both at an arbitrary source and on the entire lattice. For the leading positive eigenvalue of the operator, we obtain existence conditions determining exponential growth in the number of particles in this model.     

Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

The behavior of the maximal displacement of a supercritical branching random walk has been a subject of intense studies for a long time. But only recently the case of time-inhomogeneous branching has gained focus. The contribution of this paper is to analyze a time-inhomogeneous model with two levels of randomness. In the first step a sequence of branching laws is sampled independently according to a distribution on the set of point measures’ laws. Conditionally on the realization of this sequence (called environment) we define a branching random walk and find the asymptotic behavior of its maximal particle. It is of the form $V_n?\phi logn+o_{\mathbf{P}}(logn)$, where $V_n$ is a function of the environment that behaves as a random walk and $\phi >0$ is a deterministic constant, which turns out to be bigger than the usual logarithmic correction of the homogeneous branching random walk.