We investigate convex rearrangements, called convexifications for brevity, of stochastic processes over fixed time intervals and develop the corresponding asymptotic theory when the time intervals indefinitely expand. In particular, we obtain strong and weak limit theorems for these convexifications when the processes are Gaussian with stationary increments and then illustrate the results using fractional Brownian motion. As a theoretical basis for these investigations, we extend some known, and also obtain new, results concerning the large sample asymptotic theory for the empirical generalized Lorenz curves and the Vervaat process when observations are stationary and either short-range or long-range dependent.

We study the multiserver queue with Poisson arrivals and identical independent servers with exponentially distributed service times. Customers arriving at the system are admitted or rejected according to a fixed threshold policy. Moreover, the system is subject to holding, waiting, and rejection costs. We give a closed-form expression for the average costs and the value function for this multiserver queue. The result will then be used in a single step of policy iteration in the model where a controller has to route to several finite-buffer queues with multiple servers. We numerically show that the improved policy has a close to optimal value.

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.

We study irreducible time-homogenous Markov chains with finite state space in discrete time. We obtain results on the sensitivity of the stationary distribution and other statistical quantities with respect to perturbations of the transition matrix. We define a new closeness relation between transition matrices, and use graph-theoretic techniques, in contrast with the matrix analysis techniques previously used.

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

In this paper, we propose a customer-arrival-based insurance risk model, in which customers' potential claims are described as independent and identically distributed heavy-tailed random variables and premiums are the same for each policy. We obtain some precise large deviation results for the prospective-loss process under a mild assumption on the random index (in our case, the customer-arrival process), which is much weaker than that in the literature.

The aim of this paper is to study the distribution of colours, {Xn}, in a generalized Pólya urn model with L colours, an urn function and a random environment. In this setting, the number of actions to be taken can be greater than L, and the total number of balls added in each step can be random. The process {Xn} is expressed as a stochastic recurrent equation that fits a Robbins—Monro scheme. Since this process evolves in the (L—1)-simplex, the stability of the solutions of the ordinary differential equation associated with the Robbins—Monro scheme can be studied by means of differential algebraic equations. This approach provides a method of obtaining strong laws for the process {Xn}.

In this paper we derive the distribution of the total downtime of a repairable system during a given time interval. We allow dependence of the failure time and the repair time. The results are presented in the form of Laplace transforms which can be inverted numerically. We also discuss asymptotic properties of the total downtime.

Upper as well as lower bounds for both the large deviations and small deviations of several sup-norms associated with the displacements of a one-dimensional string driven by a Gaussian noise which is white in time and has general spatial covariance are developed.

In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝd. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.

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.

A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

We improve on previous finite time estimates for the simulated annealing algorithm which were obtained from a Cheeger-like approach. Our approach is based on a Poincaré inequality.

In this paper, we consider Galton–Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth—death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Using random walk theory, we first establish explicitly the exact distribution of the maximal partial sum of a sequence of independent and identically distributed random variables. This result allows us to obtain a new approximation of the distribution of the local score of one sequence. This approximation improves the one given by Karlin et al., which can be deduced from this new formula. We obtain a more accurate asymptotic expression with additional terms. Examples of application are given.

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.