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(d1/3) iterations to converge to stationarity.

For a particular kind of finite-server loss system in which the number and identity of servers depends on the type of the arriving call and on the state of the system, the limits of the state probabilities (as t → ∞) are found for an arbitrary service-time distribution.

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.

In this paper, we study the on-line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous-time homogeneous Markov process with a finite state space. The state of the system is hidden except for the failure state. When the system is operating, only the information obtained by condition monitoring, which is related to the working state of the system, is available. The condition monitoring observations are assumed to be in continuous range, so that no discretization is required. A recursive maximum likelihood (RML) algorithm is proposed for the on-line parameter estimation of the model. The new RML algorithm proposed in the paper is superior to other RML algorithms in the literature in that no projection is needed and no calculation of the gradient on the surface of the constraint manifolds is required. A numerical example is provided to illustrate the algorithm.

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (Xn) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Yn. Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

We study a one-dimensional supercritical branching random walk in a non-i.i.d. random environment, which considers both the branching mechanism and the step transition. This random environment is constructed using a recurrent Markov chain on a finite or countable state space. Criteria of (strong) recurrence and transience are presented for this model.

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

We consider a stochastic SIS model for the spread of an epidemic amongst a population of n individuals that are equally spaced upon the circumference of a circle. Whilst infectious, an individual, i say, makes both local and global infectious contacts at the points of homogeneous Poisson point processes. Global contacts are made uniformly at random with members of the entire population, whilst local contacts are made according to a contact distribution centred upon the infective. Individuals at the end of their infectious period return to the susceptible state and can be reinfected. The emphasis of the paper is on asymptotic results as the population size n → ∞. Therefore, a contact process with global infection is introduced representing the limiting behaviour as n → ∞ of the circle epidemics. A branching process approximation for the early stages of the epidemic is derived and the endemic equilibrium of a major outbreak is obtained. Furthermore, assuming exponential infectious periods, the probability of a major epidemic outbreak and the proportion of the population infectious in the endemic equilibrium are shown to satisfy the same equation which characterises the epidemic process.

The optimal control of dynamic models which are not completely known to the controller often requires some kind of estimation of the unknown parameters. We present conditions under which a minimum contrast estimator will be strongly consistent independently of the control used. This kind of estimator is appropriate for the adaptive or ‘estimation and control' approach in dynamic programming under uncertainty. We consider a countable-state Markov renewal model and we impose bounding and recurrence conditions of the so-called Liapunov type.

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

We consider the class of bandit problems in which each of the n ≧ 2 independent arms generates rewards according to one of the same two reward distributions, and discounting is geometric over an infinite horizon. We show that the dynamic allocation index of Gittins and Jones (1974) in this context is strictly increasing in the probability that an arm is the better of the two distributions. It follows as an immediate consequence that myopic strategies are the uniquely optimal strategies in this class of bandit problems, regardless of the value of the discount parameter or the shape of the reward distributions. Some implications of this result for bandits with Bernoulli reward distributions are given.

We analyze copulas with a nontrivial singular component by using their Markov kernel representation. In particular, we provide existence results for copulas with a prescribed singular component. The constructions not only help to deal with problems related to multivariate stochastic systems of lifetimes when joint defaults can occur with a nonzero probability, but even provide a copula maximizing the probability of joint default.

Barrieu, Rouault and Yor (2004) determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution. We determine the asymptotics of the density. This refinement can be applied to the pricing of Asian options in the Black-Scholes model.

In this paper we study a reflected AR(1) process, i.e. a process (Z n )n obeying the recursion Z n +1= max{aZ n +X n ,0}, with (X n )n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z n (in terms of transforms) in case X n can be written as Y n −B n , with (B n )n being a sequence of independent random variables which are all Exp(λ) distributed, and (Y n )n i.i.d.; when |a|<1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (B n )n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

In this paper we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes Loeffen et al. (2013).

Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper we first generalize results of Pitman and Yor (2011) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

We consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size εNN, usually called peripheral isolates in ecology, where N → ∞ and εN → 0 in such a way that εNN → ∞. The main population, as well as the colonies, independently send propagules to found new colonies (emigration), and each colony independently, eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation. We make assumptions on model parameters ensuring that the total outer population has size of the order of N and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and (only) inner lineages coalesce at constant rate per pair. This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent.