The standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has a mixing time at least of order $n^3$ , and at most of order $n^{4}$ . Bayesian inference methods rely on Markov chain Monte Carlo methods on the space of trees. Thus it is important to define good Markov chains which are easy to simulate and for which rates of convergence can be studied.

This article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster, and then we apply the developed theory to determine the limiting distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$ , where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability, and derive several exact distributional results for the all-time maximal percentage.

Recently, there has been considerable interest in the calculation of decay rates for models that can be viewed as quasi-birth-and-death (QBD) processes with infinitely many phases. In this paper we make a contribution to this endeavour by considering some classes of models in which the transition function is not homogeneous in the phase direction. We characterize the range of decay rates that are compatible with the dynamics of the process away from the boundary. In many cases, these rates can be attained by changing the transition structure of the QBD process at level 0. Our approach, which relies on the use of orthogonal polynomials, is an extension of that in Motyer and Taylor (2006) for the case where the generator has homogeneous blocks.

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Vector autoregressive processes of the first order are considered which are non-negative and optimize a linear objective function. These processes may be used in stochastic linear programming with a dynamic structure. By using Tweedie's results from the theory of Markov chains, conditions for geometric rates of convergence to stationarity (i.e. so-called geometric ergodicity) and for existence and geometric convergence of moments of these processes are obtained.

In a separate paper (Davis [2]) the characteristic functional for a replacement process due to Gani (see Gani and Yeo [1]) was obtained explicitly. The purpose of the present note is to consider the product densities for this process.

The coefficient of correlation between suitably defined statistics provides a means of testing various parametric hypotheses. The distribution of such coefficients, when the null hypothesis is not true, takes various forms in different applications. In particular, the correlation coefficients used to test hypotheses about the structural parameters in linear functional relations have doubly non-central distributions.

For tests of correlation in some typical situations, the curvature of the function defining the power in terms of the parameter in the neighbourhood of the null hypothesis is determined. This curvature is shown to be simply related to that of the function defining the squared correlation coefficient in terms of the parameter, in the neighbourhood of its minimum.

The author determines the distribution and the limit distribution of the number of partial sums greater than k (k = 0, 1, 2, …) for n mutually independent and identically distributed discrete random variables taking on the integers 1, 0, − 1, − 2, ….

This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.

This paper is concerned with the optimal number of redundant allocation to n-component coherent systems consisting of heterogeneous dependent components. We assume that the system is built up of L groups of different components, $L\geq 1$ , where there are $n_i$ components in group i, and $\sum_{i=1}^{L}n_i=n$ . The problem of interest is to allocate $v_i$ active redundant components to each component of type i, $i=1,\dots, L$ . To get the optimal values of $v_i$ we propose two cost-based criteria. One of them is introduced based on the costs of renewing the failed components and the costs of refreshing the alive ones at the system failure time. The other criterion is proposed based on the costs of replacing the system at its failure time or at a predetermined time $\tau$ , whichever occurs first. The expressions for the proposed functions are derived using the mixture representation of the system reliability function based on the notion of survival signature. We assume that a given copula function models the dependency structure between the components. In the particular case that the system is a series-parallel structure, we provide the formulas for the proposed cost-based functions. The results are discussed numerically for some specific coherent systems.

We investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework for studying the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.

The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$ , $-\infty$ , or oscillating. Whenever the Lévy process drifts to $+\infty$ , we prove that the $\kappa$ th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$ th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$ , we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.

We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

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.

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

The conditional probability that a Galton-Watson process with initial population j dies out before an independent Galton-Watson process initially i, is computed asymptotically as i or j gets large, given that both processes eventually die out. The results are interpreted qualitatively to discuss parameters of fitness for biological populations, especially with respect to the spread of mutant genes under evolutionary models.

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.