Let I be a finite set and S be a nonempty strict subset of I which is partitioned into classes, and let C(s) be the class containing s ∈ S. Let (P_s: s ∈ S) be a family of distributions on I^N, where each P_s applies to sequences starting with the symbol s. To this family, we associate a class of distributions P^(π) on I^N which depends on a probability vector π. Our main results assume that, for each s ∈ S, P_s regenerates with distribution P_s' when it encounters s' ∈ S ∖ C(s). From semiregenerative theory, we determine a simple condition on π for P^(π) to be time stationary. We give a similar result for the following more complex model. Once a symbol s' ∈ S ∖ C(s) has been encountered, there is a decision to be made: either a new region of type C(s') governed by P_s' starts or the region continues to be a C(s) region. This decision is modeled as a random event and its probability depends on s and s'. The aim in studying these kinds of models is to attain a deeper statistical understanding of bacterial DNA sequences. Here I is the set of codons and the classes (C(s): s ∈ S) identify codons that initiate similar genomic regions. In particular, there are two classes corresponding to the start and stop codons which delimit coding and noncoding regions in bacterial DNA sequences. In addition, the random decision to continue the current region or begin a new region of a different class reflects the well-known fact that not every appearance of a start codon marks the beginning of a new coding region.

In clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probability $p\gt\frac{1}{2}$. Under this design the proportion of patients in any arm converges to $\frac{1}{2}$, and the convergence rate is $n^{-1}$, as opposed to $n^{-1/2}$ under some other popular designs. The generalization of Efron's design to $K\geq2$ arms and an unequal target allocation ratio $(q_1, \ldots, q_K)$ can be found in some papers, most of which determine the allocation probabilities $p$s in a heuristic way. Nonetheless, it has been noted that by using inappropriate $p$s, the proportion of patients in the $K$ arms never converges to the target ratio. We develop a general theory to answer the question of what allocation probabilities ensure that the realized proportions under a generalized design still converge to the target ratio $(q_1, \ldots, q_K)$ with rate $n^{-1}$.

Momentum is the notion that an asset that has performed well in the past will continue to do so for some period. We study the optimal liquidation strategy for a momentum trade in a setting where the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the trader, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. Comparisons with existing results for momentum trading under incomplete information show that the assumption that the disappearance of the momentum effect is triggered by observable external shocks significantly improves the optimal strategy.

De Bruijn's identity relates two important concepts in information theory: Fisher information and differential entropy. Unlike the common practice in the literature, in this paper we consider general additive non-Gaussian noise channels where more realistically, the input signal and additive noise are not independently distributed. It is shown that, for general dependent signal and noise, the first derivative of the differential entropy is directly related to the conditional mean estimate of the input. Then, by using Gaussian and Farlie–Gumbel–Morgenstern copulas, special versions of the result are given in the respective case of additive normally distributed noise. The previous result on independent Gaussian noise channels is included as a special case. Illustrative examples are also provided.

A useful result about leftmost and rightmost paths in two-dimensional bond percolation is proved. This result was introduced without proof in Gray (1991) in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete-time contact process and two-dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.

In this paper we study the asymptotic behaviour near extinction of (sub-)critical continuous-state branching processes. In particular, we establish an analogue of Khintchine's law of the iterated logarithm near extinction time for a continuous-state branching process whose branching mechanism satisfies a given condition.

In this paper we connect Archimedean survival processes (ASPs) with the theory of Markov copulas. ASPs were introduced by Hoyle and Mengütürk (2013) to model the realized variance of two assets. We present some new properties of ASPs related to their dependency structure. We study weak and strong Markovian consistency properties of ASPs. An ASP is weak Markovian consistent, but generally not strong Markovian consistent. Our results contain necessary and sufficient conditions for an ASP to be strong Markovian consistent. These properties are closely related to the concept of Markov copulas, which is very useful in modelling different dependence phenomena. At the end we present possible applications.

We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the computer science notion of algorithm complexity. Our main result states that any weak limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the random-walk Metropolis algorithm in $d$ dimensions takes $O(d)$ iterations to converge to stationarity, while the Metropolis-adjusted Langevin algorithm takes $O(d^{1/3})$ iterations to converge to stationarity.

We consider the stability of robust scheduling policies for multiclass queueing networks. These are open networks with arbitrary routeing matrix and several disjoint groups of queues in which at most one queue can be served at a time. The arrival and potential service processes and routeing decisions at the queues are independent, stationary, and ergodic. A scheduling policy is called robust if it does not depend on the arrival and service rates nor on the routeing probabilities. A policy is called throughput-optimal if it makes the system stable whenever the parameters are such that the system can be stable. We propose two robust policies: longest-queue scheduling and a new policy called longest-dominating-queue scheduling. We show that longest-queue scheduling is throughput-optimal for two groups of two queues. We also prove the throughput-optimality of longest-dominating-queue scheduling when the network topology is acyclic, for an arbitrary number of groups and queues.

We investigate several aspects of a self-similar evolutionary process that builds a random bipolar network from building blocks that are themselves small bipolar networks. We characterize admissible outdegrees in the history of the evolution. We obtain the limit distribution of the polar degrees (when suitably scaled) characterized by its sequence of moments. We also obtain the asymptotic joint multivariate normal distribution of the number of nodes of small admissible outdegrees. Five possible substructures arise, and each has its own parameters (mean vector and covariance matrix) in the multivariate distribution. Several results are obtained by mapping bipolar networks into Pólya urns.

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The environment is cast in terms of a Jackson/Gordon–Newell network although alternative interpretations are possible. The main tool is the detailed balance equations. The difference compared to earlier works is that the position of the random walk influences the transition intensities of the network environment and vice versa, creating strong correlations. The form of the stationary distribution is closely related to the well-known product formula.

We consider a directed graph on the integers with a directed edge from vertex $i$ to $j$ present with probability $n^{-1}$, whenever $i\lt j$, independently of all other edges. Moreover, to each edge $(i,j)$ we assign weight $n^{-1}(j-i)$. We show that the closure of vertex $0$ in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as $n\rightarrow \infty$. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most $\rho$ from the root.

We consider three different schemes for signal routeing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with independent and identically distributed weights representing the strength of the transceivers. The edges of the tree are also equipped with independent and identically distributed weights, representing the costs for passing the edges. For each one of our schemes, we derive sharp conditions on the distributions of the vertex weights and the edge weights that determine when the root can transmit a signal over arbitrarily large distances.

During the course of a day an individual typically mixes with different groups of individuals. Epidemic models incorporating population structure with individuals being able to infect different groups of individuals have received extensive attention in the literature. However, almost exclusively the models assume that individuals are able to simultaneously infect members of all groups, whereas in reality individuals will typically only be able to infect members of any group they currently reside in. In this paper we develop a model where individuals move between a community and their household during the course of the day, only infecting within their current group. By defining a novel branching process approximation with an explicit expression for the probability generating function of the offspring distribution, we are able to derive the probability of a major epidemic outbreak.

The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as a continuous-time Markov chain model describing the genealogical relationship among sampled genes from a panmictic population of a species. The random mating in a population is a strict condition and the genealogical structure of the population has a strong influence on the genetic variability and the evolution of the species. In this paper, starting from a discrete-time Markov chain model, we show the weak convergence to a continuous-time Markov chain, called the structured coalescent model, describing the genealogy of the sampled genes from whole population by means of passing the limit of the population size. Herbots (1997) proved the weak convergence to the structured coalescent on the condition of conservative migration and Wright–Fisher-type reproduction. We will give the proof on the condition of general migration rates and exchangeable reproduction.

Respondent-driven sampling (RDS) is frequently used when sampling from hidden populations. In RDS, sampled individuals pass on participation coupons to at most $c$ of their acquaintances in the community ($c=3$ being a common choice). If these individuals choose to participate, they in turn pass coupons on to their acquaintances, and so on. The process of recruiting is shown to behave like a new Reed–Frost-type network epidemic, in which `becoming infected' corresponds to study participation. We calculate $R_0$, the probability of a major `outbreak', and the relative size of a major outbreak for $c\lt\infty$ in the limit of infinite population size and compare to the standard Reed–Frost epidemic. Our results indicate that $c$ should often be chosen larger than in current practice.

We present an asymptotically optimal importance sampling for Monte Carlo simulation of the Laplace transform of exponential Brownian functionals which plays a prominent role in many disciplines. To this end we utilize the theory of large deviations to reduce finding an asymptotically optimal importance sampling measure to solving a calculus of variations problem. Closed-form solutions are obtained. In addition we also present a path to the test of regularity of optimal drift which is an issue in implementing the proposed method. The performance analysis of the method is provided through the Dothan bond pricing model.

The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

In this paper we investigate the functional central limit theorem (CLT) for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional CLT is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.

It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in $(-\infty,0]$, say $\sigma$, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on $[0,\sigma]$. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

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 $\infty$ and the mutation probability goes to 0, the asymptotic relative frequency of the sequences differing on $k$ digits from the master sequence approaches $(\sigma{\rm e}^{-a}-1)({a^k}/{k!})\sum_{i\geq1}{i^k}/{\sigma^i},$ where $\sigma$ 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 $\mathcal{Q}(\sigma,a)$ on the nonnegative integers given by the above equation is the quasispecies distribution with parameters $\sigma$ and $a$.

In this paper we study a special class of size dependent branching processes. We assume that for some positive integer $K$ as long as the population size does not exceed level $K$, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level $K$, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time $T$. The question of when the mean value $\mathbb{E}(T)$ is finite or infinite is also addressed.

In this paper we study a generalized coupon collector problem, which consists of analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the nonnull coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one which stochastically maximizes the time needed to collect a fixed number of distinct coupons.

This note is motivated by Blom's work in 1989. We consider a generalized Ehrenfest urn model in which a randomly-chosen ball has a positive probability of moving from one urn to the other urn. We use recursion relations between the mean transition times to derive formulas in terms of finite sums, which are shown to be equivalent to the definite integrals obtained by Blom.