We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 < α < 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

Taylor's law (TL) originated as an empirical pattern in ecology. In many sets of samples of population density, the variance of each sample was approximately proportional to a power of the mean of that sample. In a family of nonnegative random variables, TL asserts that the population variance is proportional to a power of the population mean. TL, sometimes called fluctuation scaling, holds widely in physics, ecology, finance, demography, epidemiology, and other sciences, and characterizes many classical probability distributions and stochastic processes such as branching processes and birth-and-death processes. We demonstrate analytically for the first time that a version of TL holds for a class of distributions with infinite mean. These distributions, a subset of stable laws, and the associated TL differ qualitatively from those of light-tailed distributions. Our results employ and contribute to the methodology of Albrecher and Teugels (2006) and Albrecher et al. (2010). This work opens a new domain of investigation for generalizations of TL.

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=𝔼[Z1]. We assume that we are in the Schröder case with 𝔼Z1 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→∞.

The explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

In this paper we derive limit theorems for the conditional distribution of X1 given Sn=sn as n→ ∞, where the Xi are independent and identically distributed (i.i.d.) random variables, Sn=X1+··· +Xn, and sn/n converges or sn ≡ s is constant. We obtain convergence in total variation of PX1∣ Sn/n=s to a distribution associated to that of X1 and of PnX1∣ Sn=s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

We present a method for computing the probability density function (PDF) and the cumulative distribution function (CDF) of a nonnegative infinitely divisible random variable X. Our method uses the Lévy-Khintchine representation of the Laplace transform Ee-λX = e-ϕ(λ), where ϕ is the Laplace exponent. We apply the Post-Widder method for Laplace transform inversion combined with a sequence convergence accelerator to obtain accurate results. We demonstrate this technique on several examples, including the stable distribution, mixtures thereof, and integrals with respect to nonnegative Lévy processes.

In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

Let T = (T1, T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑j=1∞TjWj, where W, W1, W2,… have distribution μ and T, W1, W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions for Tj ≥ 0: essentially under the condition that E ∑j=1∞Tj2 log+Tj2 < ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

We study certain stochastic processes arising in probabilistic modelling. We discuss the limit behavior of these processes and estimate the rate of convergence to the limit.

Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn, and independent copies A1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.