For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} <\infty$ for $\delta \ge 0$ , where $0 \mathbb{Z}^d\,:\!=\,\mathbb{R}^d$ , and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for Pickands constants $H_{Z}^\delta$ , defined for $Z(t)= \exp( \sqrt{ 2} B_\alpha (t)- \lvert {t} \rvert^{2\alpha })$ , $t\in \mathbb{R}$ , with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$ . The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants.

We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ s ≤ t} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τy and Mt: for example, if m(y) = Eτy is of rapid variation then Mt / m-1(t) →w 1 as t → ∞, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

We derive multivariate Sobel–Uppuluri–Galambos-type lower bounds for the probability that at least a1 and at least a2, and for the probability that exactly a1 and a2, out of n and N events, occur. The lower bound presented here reduces to a sharper bound than that of Galambos and Lee (1992). Our approach is by way of indicator functions and bivariate binomial moments. A new concept, marginal Bonferroni summation, is introduced in this paper.

For a sequence of finite state space birth–death processes, each having a single absorbing state, we show that, under certain conditions, as the size of the state space tends to infinity, the quasistationary distributions converge to the stationary distribution of a limiting infinite state space birth–death process. This generalizes a result of Keilson and Ramaswamy by allowing birth and death rates to depend upon the size of the state space. We give sufficient conditions under which the convergence result of Keilson and Ramaswamy remains valid. The generalization allows us to apply our convergence result to examples from population biology: a Pearl–Verhulst logistic population growth model and the susceptible-infective-susceptible (SIS) model for infectious spread. The limit distributions obtained suggest new finite-population approximations to the quasistationary distributions of these models, obtained by the method of cumulant closure. The new approximations are found to be both simple in form and accurate.

This paper analyzes players’ long-run behavior in an evolutionary model with time-varying mutations under both uniform and local interaction rules. It is shown that a risk-dominant Nash equilibrium in a 2 × 2 coordination game would emerge as the long-run equilibrium if and only if mutation rates do not decrease to zero too fast under both interaction methods. The convergence rates of the dynamic system under both interaction rules are also derived. We find that the dynamic system with local matching may not converge faster than that with uniform matching.

Points randomly placed in Rn are independently assigned a speed and direction of motion. The time between collisions with a central n-sphere is obtained. If instead of straight-line motion the points perform Brownian motion, the analogous inter-collision-time distribution is obtained.

We present a new approach to the processor-sharing queue that allows us to study the accumulated work process of a job that requires an amount of processing time x. Our approach simplifies the proofs of some earlier results on expected conditional response times and extends them to the M/G/φ (·) class. The approach illuminates some of the paradoxical features of these systems.

We consider a variant of the house-selling problem in which the seller has n objects for sale and the buyers come sequentially one by one. The objective of this paper is to find the seller's optimal strategy for choosing his selling price so as to maximize the average reward per selling attempt until the nth house is sold.

Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, for r-out-of-n systems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering.

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is waiting at the polled queue. Naive intuition would lead one to believe that the SD version should perform better than the SI version. We characterize the difference in the expected waiting times of these two versions, and we uncover some surprising facts. In particular, we show that, regardless of the server utilization or the service-time distribution, the SD version performs (i) the same as, (ii) worse than, or (iii) better than its SI counterpart if the switchover and setup times are, respectively, (i) both constants, (ii) variable (i.e. non-deterministic) and constant, or (iii) constant and variable. Only (iii) is consistent with naive intuition.

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically, it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

Regenerative stopping problems are stopping problems which recommence from the initial state upon stopping. An algorithm is presented which solves semi-Markov regenerative stopping problem with a finite number of continue actions by solving a sequence of stopping problems. New results for the optimal stopping problem are obtained as well as for the regenerative stopping problem. A model in the literature is used as a detailed example of the algorithm.

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.

We compare dependence in stochastically monotone Markov processes with partially ordered Polish state spaces using the concordance and supermodular orders. We show necessary and sufficient conditions for the concordance order to hold both in terms of the one-step transition probabilities for discrete-time processes and in terms of the corresponding infinitesimal generators for continuous-time processes. We give examples showing that a stochastic monotonicity assumption is not necessary for such orderings. We indicate relations between dependence orderings and, variously, the asymptotic variance-reduction effect in Monte Carlo Markov chains, Cheeger constants, and positive dependence for Markov processes.

Based on a semi-Markov process J(t), t ≧ 0, a reward process Z(t), t ≧ 0, is introduced where it is assumed that the reward function, p(k, x) is nonlinear; if the reward function is linear, i.e. ρ (k, x) = kx, the reward process Z(t), t ≧ 0, becomes the classical one, which has been considered by many authors. An explicit formula for E(Z(t)) is given in terms of the moments of the sojourn time distribution at t, when the reward function is a polynomial.

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.