There are standard modifications of certain compactly supported wavelets that yield orthonormal bases on a bounded interval. We extend one such construction to those wavelets, such as ‘coiflets', that may have fewer vanishing moments than had to be assumed previously. Our motivation lies in function estimation in statistics. We use these boundary-modified coiflets to show that the discrete wavelet transform of finite data from sampled regression models asymptotically provides a close approximation to the wavelet transform of the continuous Gaussian white noise model. In particular, estimation errors in the discrete setting of computational practice need not be essentially larger than those expected in the continuous setting of statistical theory.

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

For independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

In this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.

Block, Savits and Singh previously presented a preservation result for bathtub-shaped functions. In this paper, we strengthen and extend their results to a class of functions which allow for more than one change of monotonicity. Various applications are also considered.

We consider a model for the spread of an epidemic through a population divided into m groups, in which infectives move from group to group and infect only within their current group. For both deterministic and stochastic versions of this model, the effect on the total size of the epidemic of varying the speed with which infectives move between groups is considered. We also compare the distribution of the total size of this model with that of a suitably matched model in which infectives cannot move between groups, but are able to infect outside their own group.

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.

We give a sample path proof of the well-known Wiener–Hopf identity F = G– + G+ – G– ∗G+ which relates the ladder-height distributions G– and G+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.

The stationary distribution for the population frequencies under an infinite alleles model is described as a random sequence (x1, x2, · ··) such that Σxi = 1. Likelihood ratio theory is developed for random samples drawn from such populations. As a result of the theory, it is shown that any parameter distinguishing an infinite alleles model with selection from the neutral infinite alleles model cannot be consistently estimated based on gene frequencies at a single locus. Furthermore, the likelihood ratio (neutral versus selection) converges to a non-trivial random variable under both hypotheses. This shows that if one wishes to test a completely specified infinite alleles model with selection against neutrality, the test will not obtain power 1 in the limit.

Determinantal point processes (DPPs) enable the modeling of repulsion: they provide diverse sets of points. The repulsion is encoded in a kernel K that can be seen, in a discrete setting, as a matrix storing the similarity between points. The main exact algorithm to sample DPPs uses the spectral decomposition of K, a computation that becomes costly when dealing with a high number of points. Here we present an alternative exact algorithm to sample in discrete spaces that avoids the eigenvalues and the eigenvectors computation. The method used here is innovative, and numerical experiments show competitive results with respect to the initial algorithm.

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 → ∞.

We consider a class of statistics connected with the history of a Markov chain before the first occurrence of certain runs. Methods for deriving generating functions, distributions and moments of the statistics considered are given. In the case of i.i.d. random variables, explicit formulas are derived.

This note indicates two alternative proofs for a result of Galambos and Seneta [1] on the distribution of ratios of successive record times in a sequence of independent and identically distributed random variables. Some consequences of one method of proof are discussed, and its analogues in a continuous time setting are described.

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dt as a measure of the speed of convergence towards stationarity of the process {X(t), t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t), t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.

We consider, in discrete time, a single machine system that operates for a period of time represented by a general distribution. This machine is subject to failures during operations and the occurrence of these failures depends on how many times the machine has previously failed. Some failures are repairable and the repair times may or may not depend on the number of times the machine was previously repaired. Repair times also have a general distribution. The operating times of the machine depend on how many times it has failed and was subjected to repairs. Secondly, when the machine experiences a nonrepairable failure, it is replaced by another machine. The replacement machine may be new or a refurbished one. After the Nth failure, the machine is automatically replaced with a new one. We present a detailed analysis of special cases of this system, and we obtain the stationary distribution of the system and the optimal time for replacing the machine with a new one.

A stochastic version of the logistic model for population growth is considered, and the general form of an optimal policy is found for hunting the population so as to maximise the long-term average number of captures per unit time. This optimal policy is described by a critical population size x∗such that it is optimal to hunt if and only if the population size is greater than or equal to x∗. Methods of determining x∗for given parameter values are provided, and some properties of the optimal policy as the population size tends to infinity are proved.

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

We consider a finite-capacity queueing system where arriving customers offer rewards which are paid upon acceptance into the system. The gatekeeper, whose objective is to ‘maximize’ rewards, decides if the reward offered is sufficient to accept or reject the arriving customer. Suppose the arrival rates, service rates, and system capacity are changing over time in a known manner. We show that all bias optimal (a refinement of long-run average reward optimal) policies are of threshold form. Furthermore, we give sufficient conditions for the bias optimal policy to be monotonic in time. We show, via a counterexample, that if these conditions are violated, the optimal policy may not be monotonic in time or of threshold form.

An optimal linear control (4) is deduced for the symmetric team control problem specified by relations (1)–(3). The generating function (6) for the optimal operator is in general irrational if there is more than one agent (controller), indicating that the recursions associated with effective state estimation are no longer of finite order, as in the single-agent case.

Some sufficient conditions for non-ergodicity are given for a Markov chain with denumerable state space. These conditions generalize Foster's results, in that unbounded Lyapunov functions are considered. Our criteria directly extend the conditions obtained in Kaplan (1979), in the sense that a class of Lyapunov functions is studied. Applications are presented through some examples; in particular, sufficient conditions for non-ergodicity of a multidimensional Markov chain are given.