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In a reinforced Galton–Watson process with reproduction law ν$$ \boldsymbol{\nu} $$ and memory parameter q∈(0,1)$$ q\in \left(0,1\right) $$, the number of children of a typical individual either, with probability q$$ q $$, repeats that of one of its forebears picked uniformly at random, or, with complementary probability 1−q$$ 1-q $$, is given by an independent sample from ν$$ \boldsymbol{\nu} $$. We estimate the average size of the population at a large generation, and in particular, we determine explicitly the Malthusian growth rate in terms of ν$$ \boldsymbol{\nu} $$ and q$$ q $$. Our approach via the analysis of transport equations owes much to works by Flajolet and co‐authors. [ABSTRACT FROM AUTHOR]

For the density of Galton-Watson processes in the Schröder case, we derive a complete left tail asymptotic series consisting of power terms multiplied by periodic factors. [ABSTRACT FROM AUTHOR]

Consider a supercritical Galton–Watson process with immigration (X n ; n ≥ 0) . The Lotka–Nagaev estimator X n + 1 X n is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals. [ABSTRACT FROM AUTHOR]

We consider branching random processes with immigration starting from random number of items. In this work we provide estimations from above for the moments of such processes. [ABSTRACT FROM AUTHOR]

Consider a supercritical Galton–Watson process with immigration (Xn;n≥0). The Lotka–Nagaev estimator Xn+1Xn is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals.

Cross‐linking and branching of primary polymer molecules are investigated using the Galton–Watson (GW) process. Starting with the probability generating function (pgf) of the primary molecular weight distribution (MWD), analytical expressions are derived for the bivariate pgfs g(nbr, s) of branched polymers which depend also on the number of branch points nbr. The bivariate MWDs n(nbr, i) (i: number of molecular units) are then derived as Taylor expansions in s. All three cases of random branching: X‐shaped (cross‐linking), T‐shaped (only one end takes part in the branching process), and H‐shaped (both ends can take part in the branching process) are treated. An extension of the formalism does not require the construction of the pgf and allows the direct use of the MWD of the primary chains. However, using pgfs allows to go past the gel point and to determine the MWD and content of the sol. Explicit expressions are given for special distributions: the mono modal, the most probable, the Schulz‐Zimm, the Poisson, and the Catalan distribution for the cases of X‐shaped and T‐shaped branching. [ABSTRACT FROM AUTHOR]

Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Peköz et al. (2016) exploited characterizing distributional fixed point equations to obtain uniform error bounds for generalized gamma approximations using Stein's method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Stein's method. Applications are given to concentration inequalities for preferential attachment random graphs, branching processes, random walk local time statistics and the size of random subtrees of uniformly random binary rooted plane trees. [ABSTRACT FROM AUTHOR]

Abstract–We use a connection between compositional kernels and branching processes via Mehler's formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation functions in compositional neural networks. We study the unscaled and rescaled limits of the compositional kernels and explore the different phases of the limiting behavior, as the compositional depth increases. We investigate the memorization capacity of the compositional kernels and neural networks by characterizing the interplay among compositional depth, sample size, dimensionality, and nonlinearity of the activation. Explicit formulas on the eigenvalues of the compositional kernel are provided, which quantify the complexity of the corresponding reproducing kernel Hilbert space. On the methodological front, we propose a new random features algorithm, which compresses the compositional layers by devising a new activation function. for this article are available online. [ABSTRACT FROM AUTHOR]

Critical branching processes in a varying environment behave much the same as critical Galton–Watson processes. In this note we like to confirm this finding with regard to the underlying genealogical structures. In particular, we consider the most recent common ancestor given survival and the corresponding reduced branching processes, in the spirit of Zubkov (1975) and Fleischmann and Siegmund-Schultze (1977). [ABSTRACT FROM AUTHOR]

We discuss how to recover a secret bitstring given partial information obtained during a computation over that string, assuming the computation is a deterministic algorithm processing the secret bits sequentially. That abstract situation models certain types of side-channel attacks against discrete logarithm and RSA-based cryptosystems, where the adversary obtains information not on the secret exponent directly, but instead on the group or ring element that varies at each step of the exponentiation algorithm.

Birth‐and‐death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, as the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. A further difficulty arises when the discrete observations are not equi‐spaced, for example, when census data are unavailable for some years. We present two approaches to estimating the birth, death, and growth rates of a discretely observed linear birth‐and‐death process: via an embedded Galton‐Watson process and by maximizing a saddlepoint approximation to the likelihood. We study asymptotic properties of the estimators, compare them on numerical examples, and apply the methodology to data on monitored populations. [ABSTRACT FROM AUTHOR]

We introduce generalized dynamical pruning on rooted binary trees with edge lengths that encompasses a number of discrete and continuous pruning operations, including the tree erasure and Horton pruning. The pruning removes parts of a tree T, starting from the leaves, according to a pruning function defined on descendant subtrees within T. We prove the invariance of critical binary Galton–Watson tree with exponential edge lengths with respect to the generalized dynamical pruning for an arbitrary admissible pruning function. These results facilitate analysis of the continuum 1-D ballistic annihilation model A + A → ∅ for a constant particle density and initial velocity that alternates between the values of ± 1 . We show that the model's shock wave is isometric to the level set tree of the potential function, and the model evolution is equivalent to the generalized dynamical pruning of the shock wave tree. [ABSTRACT FROM AUTHOR]

We consider Branching Random Processes with Immigration starting from random number of items. In this work we provide estimations from above for the moments of such processes. [ABSTRACT FROM AUTHOR]

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class of positive integer-valued distributions with power tails. [ABSTRACT FROM AUTHOR]

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$ , properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$ , conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes. [ABSTRACT FROM AUTHOR]

The class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions. [ABSTRACT FROM AUTHOR]

In France, every town has a memorial to its inhabitants who perished in the First World War. Numbering in the hundreds of thousands, the names of these individuals were engraved on such monuments to ensure their permanence in the nation's history. However, some names have disappeared with the deaths of their bearers. This article estimates the extent of the disappearance of surnames attributable to the war based on a comparison of two databases: the 'Morts pour la France' database and the INSEE surname file. How many surnames have disappeared? Which regions were most affected?. [ABSTRACT FROM AUTHOR]

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R 0 , the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R 0 , the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k = 1 , 2 , ... , M). In the case of viruses, however, if R 0 is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton–Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman–Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R 0. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R 0 may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups. [ABSTRACT FROM AUTHOR]

The basis of this paper is the elementary observation that the n -step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs become linear and purely dilatational when expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence of universal critical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generate all possible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix. [ABSTRACT FROM AUTHOR]

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton-Watson branching process with immigration is described. We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well. [ABSTRACT FROM AUTHOR]

We study the porosity properties of fractal percolation sets $$E\subset \mathbb {R}^d$$ . Among other things, for all $$0<\varepsilon <\tfrac{1}{2}$$ , we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than $$\tfrac{1}{2}-\varepsilon $$ , or the lower porosity is larger than $$\varepsilon $$ . Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton-Watson process. [ABSTRACT FROM AUTHOR]

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten- Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the nth generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of (J. Stat. Phys. 143 (3) (2011) 420-446) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165-170) in our framework. [ABSTRACT FROM AUTHOR]

Branching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality. [ABSTRACT FROM PUBLISHER]

Given a supercritical Galton‒Watson process {Zn} and a positive sequence {εn}, we study the limiting behaviors of ℙ(SZn/Zn≥εn) with sums Sn of independent and identically distributed random variables Xi and m=피[Z1]. We assume that we are in the Schröder case with 피Z1 log Z1<∞ and X1 is in the domain of attraction of an α-stable law with 0<α<2. As a by-product, when Z1 is subexponentially distributed, we further obtain the convergence rate of Zn+1/Zn to m as n→∞. [ABSTRACT FROM AUTHOR]

We consider a discrete-time host–parasite model for a population of cells which are colonized by proliferating parasites. The cell population grows like an ordinary Galton–Watson process, but in reflection of real biological settings the multiplication mechanisms of cells and parasites are allowed to obey some dependence structure. More precisely, the number of offspring produced by a mother cell determines the reproduction law of a parasite living in this cell and also the way the parasite offspring is shared into the daughter cells. In this article, we provide a formal introduction of this branching-within-branching model and then focus on the property of parasite extinction. We establish equivalent conditions for almost sure extinction of parasites and find a strong relation of this event to the behavior of parasite multiplication along a randomly chosen cell line through the cell tree, which forms a branching process in random environment. We then focus on asymptotic results for relevant processes in the case when parasites survive. In particular, limit theorems for the processes of contaminated cells and of parasites are established by using martingale theory and the technique of size-biasing. The results for both processes are of Kesten–Stigum type by including equivalent integrability conditions for the martingale limits to be positive with positive probability. The case when these conditions fail is also studied. For the process of contaminated cells, we show that a proper Heyde–Seneta norming exists such that the limit is nondegenerate. [ABSTRACT FROM AUTHOR]

We study a classical multitype Galton-Watson process with mutation and selection. The individuals are sequences of fixed length over a finite alphabet. On the sharp peak fitness landscape together with independent mutations per locus, we show that, as the length of the sequences goes to oo and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on k digits from the master sequence approaches (σe-a- l)(ak/k!)Σi≥1ik/σi, where σ is the selective advantage of the master sequence and a is the product of the length of the chains with the mutation probability. The probability distribution Q(σ, a) on the nonnegative integers given by the above equation is the quasispecies distribution with parameters σ and a. [ABSTRACT FROM AUTHOR]

To understand human population dynamics fully, before considering complex human agency it may be useful to construct baseline models to see where such agency may and may not be necessary. In fact, the dynamics of human populations may be amenable to mathematical modeling with relatively parsimonious mechanisms. We review some of the more prominent of such models, namely, the spatial Galton-Watson (GW) model, modifications of the GW model that add migration and immigration, and the Bolker-Pacala model, in which mortality (or birth rate) is affected by competition. We show that change in the distribution of population density over the last century for 12 American rural states may be captured by the simplest of the models, the spatial GW model. [ABSTRACT FROM PUBLISHER]

We characterize the sets of offspring laws on which the Lotka–Nagaev estimator for the mean of a supercritical Galton–Watson process is qualitatively robust. These are exactly the locally uniformly integrating sets of offspring laws, which may be quite large. If the corresponding global property is assumed instead, we obtain uniform robustness as well. We illustrate both results with a number of concrete examples. As a by-product of the proof we obtain that the Lotka–Nagaev estimator is [locally] uniformly weakly consistent on the respective sets of offspring laws, conditionally on non-extinction. [ABSTRACT FROM AUTHOR]

Probabilistic inequalities are deduced for all values of a breeding coefficient A of a branching process. Previously, only the case A = 1 was considered. Separately the cases are studied such that the Cramer condition is fulfilled and there exists only the finite number of moments. Thereafter one of the obtained inequalities is applied for deducing the Rosenthal-type moment inequality for A ≧ 1. [ABSTRACT FROM AUTHOR]

A synthesis of the practical theoretical main results is presented that involves the analysis of the process of Galton-Watson; as they are the results asymptotics for the cases critical subcritical and supercritical, the time of extinction of the process, the estimate of the reproduction average way maximum likelihood and the construction of some random variables which were simulated to verify their behavior normal asymptotically.Se presenta una síntesis de las principales características que se incluyen al realizar un análisis del proceso de Galton-Watson: el tiempo de extinción del proceso, los resultados asintóticos para los casos crítico, subcrítico y supercrítico, la estimación por máxima verosimilitud del promedio de reproducción y la construcción de algunas variables aleatorias simuladas para verificar su comportamiento normal asintótico.

We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton–Watson sub-trees of the Lévy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate λ , see Duquesne and Le Gall (2002). The tree-valued process, as λ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters: the pruning parameter θ and the intensity λ . Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as θ increases) and description of the growing process (backward in time that is as θ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical Lévy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton–Watson trees studied by Abraham, Delmas and He (2012). [ABSTRACT FROM AUTHOR]

Mathematical models of a cell invasion wave have included both continuum partial differential equation (PDE) approaches and discrete agent-based cellular automata (CA) approaches. Here we are interested in modelling the spatial and temporal dynamics of the number of divisions (generation number) that cells have undergone by any time point within an invasion wave. In the CA framework this is performed from agent lineage tracings, while in the PDE approach a multi-species generalized Fisher equation is derived for the cell density within each generation. Both paradigms exhibit qualitatively similar cell generation densities that are spatially organized, with agents of low generation number rapidly attaining a steady state (with average generation number increasing linearly with distance) behind the moving wave and with evolving high generation number at the wavefront. This regularity in the generation spatial distributions is in contrast to the highly stochastic nature of the underlying lineage dynamics of the population. In addition, we construct a method for determining the lineage tracings of all agents without labelling and tracking the agents, but through either a knowledge of the spatial distribution of the generations or the number of agents in each generation. This involves determining generation-dependent proliferation probabilities and using these to define a generation-dependent Galton–Watson (GDGW) process. Monte-Carlo simulations of the GDGW process are used to determine the individual lineage tracings. The lineages of the GDGW process are analyzed using Lorenz curves and found to be similar to outcomes generated by direct lineage tracing in CA realizations. This analysis provides the basis for a potentially useful technique for deducing cell lineage data when imaging every cell is not feasible. [ABSTRACT FROM AUTHOR]

Upper bounds for the expected time to extinction in the Galton-Watson process are obtained. We also found upper and lower bounds for the probability of extinction of this process. These bounds improve some bounds previously obtained by other authors. [ABSTRACT FROM AUTHOR]