A continuous-state branching process with immigration having branching mechanism $\Psi$ and immigration mechanism $\Phi$ , a CBI $(\Psi,\Phi)$ process for short, may have either of two different asymptotic regimes, depending on whether $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ or $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ . When $\int_{0}\frac{\Phi(u)}{|\Psi(u)|}\textrm{d} u<\infty$ , the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When $\scriptstyle\int_{0}\tfrac{\Phi(u)}{|\Psi(u)|}\textrm{d} u=\infty$ , immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).

This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$ matrices. In particular, for conservative and subcritical affine processes we show that a finite $\log$ -moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: first, in a specific metric induced by the Laplace transform, and second, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter.

In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.

We show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n − 1) · · · X(1) of independent and identically distributed copies X(k) of X converge almost surely as n → ∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

In this paper, following an open portfolio approach, we show how to estimate a Bonus-malus system evolution.

Considering a model for the number of new annual policies, we obtain ML estimators, asymptotic distributions and confidence regions for the expected number of new policies entering the portfolio in each year, as well as for the expected number and proportion of insureds in each bonus class, by year of enrollment. Confidence regions for the distribution of policyholders result in confidence regions for optimal bonus scales.

Our treatment is illustrated by an example with numerical results.

When the search algorithm QuickSelect compares keys during its execution in order to find a key of target rank, it must operate on the keys' representations or internal structures, which were ignored by the previous studies that quantified the execution cost for the algorithm in terms of the number of required key comparisons. In this paper we analyze running costs for the algorithm that take into account not only the number of key comparisons, but also the cost of each key comparison. We suppose that keys are represented as sequences of symbols generated by various probabilistic sources and that QuickSelect operates on individual symbols in order to find the target key. We identify limiting distributions for the costs, and derive integral and series expressions for the expectations of the limiting distributions. These expressions are used to recapture previously obtained results on the number of key comparisons required by the algorithm.

We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

Consider an infinite-source marked Poisson process to model end user inputs to a data network. At Poisson times, connections are initated. The connection is characterized by a triple (F, L, R) denoting the total quantity of transmitted data in a connection, the length or duration of the connection, and the transmission rate; the three quantities are related by F = LR. How critical is the dependence structure of the mark for network characteristics such as burstiness, distribution tails of cumulative input, and long-range dependence properties of traffic measured in consecutive time slots? In a previous publication (D'Auria and Resnick (2006)) we assumed that F and R were independent. Here we assume that L and R are independent. The change in dependence assumptions means that the model properties change dramatically: tails of cumulative input per time slot are dramatically heavier, traffic cannot be approximated by a Gaussian distribution, and the decay of dependence cannot be measured in the traditional way using correlation functions. Different network applications are likely to have different mark dependence structure. We argue that the present independence assumption on L and R is likely to be appropriate for network applications such as streaming media or peer-to-peer networks. Our conclusion is that it is desirable to separate network traffic by application and to model each application with its own appropriate dependence structure.

We consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that f(p) and f(1/p) ∈ {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and Halász's and Ruzsa's inequalities are used. We present a few examples of application of the given results to some sets of fractions.

We review characteristics of data traffic which we term stylized facts: burstiness, long-range dependence, heavy tails, bursty behavior determined by high-bandwidth users, and dependence determined by users without high transmission rates. We propose an infinite-source Poisson input model which supplies traffic in adjacent time slots. We study properties of the model as slot width decreases and traffic intensity increases. This model has the ability to account for many of the stylized facts.

A continuous-state population-size-dependent branching process {Xt} is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt / t obtained when Xt tends to infinity.

We generalize a theorem due to Keilson on the approximate exponentiality of waiting times for rare events in regenerative processes. We use the result to investigate the limit distribution for a family of first entrance times in a sequence of Ehrenfest urn models. As a second application, we consider approximate pattern matching, a problem arising in molecular biology and other areas.

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

Consider an inhomogeneous Poisson process X on [0, T] whose unknown intensity function ‘switches' from a lower function g∗ to an upper function h∗ at some unknown point θ∗. What is known are continuous bounding functions g and h such that g∗(t) ≤ g(t) ≤ h(t) ≤ h∗(t) for 0 ≤ t ≤ T. It is shown that on the basis of n observations of the process X the maximum likelihood estimate of θ∗ is consistent for n →∞, and also that converges in law and in pth moment to limits described in terms of the unknown functions g∗ and h∗.

An important model in communications is the stochastic FM signal st = A cos , where the message process {mt} is a stochastic process. In this paper, we investigate the linear models and limit distributions of FM signals. Firstly, we show that this non-linear model in the frequency domain can be converted to an ARMA (2, q + 1) model in the time domain when {mt} is a Gaussian MA (q) sequence. The spectral density of {St} can then be solved easily for MA message processes. Also, an error bound is given for an ARMA approximation for more general message processes. Secondly, we show that {St} is asymptotically strictly stationary if {mt} is a Markov chain satisfying a certain condition on its transition kernel. Also, we find the limit distribution of st for some message processes {mt}. These results show that a joint method of probability theory, linear and non-linear time series analysis can yield fruitful results. They also have significance for FM modulation and demodulation in communications.

Denote by Sn the set of all distinct rooted trees with n labeled vertices. Define τn as the total height of a tree chosen at random in the set Sn, assuming that all the possible nn–1 choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that is, the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn and their asymptotic behavior as n → ∞.

Let {Xk} be an i.i.d. sequence taking values in ℝ2 with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of {X1, · ··, Xn} is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.

This paper is concerned with the Markovian sequence Xn = Zn max{Xn–1, Yn},n ≧ 1, where X0 is any random variable, {Zn} and {Yn} are independent sequences of i.i.d. random variables both independent of X0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn's are random variables concentrated on the interval [0, 1], namely having a beta distribution.

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.