In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ > 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn, n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn, n ≥ 0} is a sequence of independent and identically distributed random variables.

We consider a nearly unstable, or near unit root, AR(1) process with regularly varying innovations. Two different approximations for the stationary distribution of such processes exist: a Gaussian approximation arising from the nearly unstable nature of the process and a heavy-tail approximation related to the tail asymptotics of the innovations. We combine these two approximations to obtain a new uniform approximation that is valid on the entire real line. As a corollary, we obtain a precise description of the regions where each of the Gaussian and heavy-tail approximations should be used.

In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.

We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p. We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑jpj2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c−1, under two constraints on p that exclude a thin set of distributions p. One of the constraints is c = o(ln−2n). This ln−2n is shown to be a threshold value: for c = ω(ln−2n), there exists p with c(p) = c such that the expected coalescence time far exceeds c−1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

It is well known that the law of a Brownian motion started from x > 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.

We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the interarrival and/or service times are bounded, and the other is when they become large. Both require a heavy-traffic condition. Tail asymptotics and heavy-traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.

The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained.

We consider M/G/1 queues with exhaustive service and generalized vacations, where at the end of every busy period the server either follows a mixed vacation policy from a given vacation policy set or stays idle. A simple recursive formula for the moments of the stationary waiting time is provided. This formula results in the decomposition property for our model immediately. It also enables us to derive many existing results for the M/G/1 queues with various vacation policies.

We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction cost. We prove that in the continuous-time Black‒Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.

In this paper we consider linear functions constructed on two different weighted branching processes and provide explicit bounds for their Kantorovich–Rubinstein distance in terms of couplings of their corresponding generic branching vectors. Motivated by applications to the analysis of random graphs, we also consider a variation of the weighted branching process where the generic branching vector has a different dependence structure from the usual one. By applying the bounds to sequences of weighted branching processes, we derive sufficient conditions for the convergence in the Kantorovich–Rubinstein distance of linear functions. We focus on the case where the limits are endogenous fixed points of suitable smoothing transformations.

A learning matrix is defined by a set of input and output pattern vectors. The entries in these vectors are zeros and ones. The matrix is the maximum of the outer products of the input and output pattern vectors. The entries in the matrix are also zeros and ones. The product of this matrix with a selected input pattern vector defines an activity vector. It is shown that when the patterns are taken to be random, then there are central limit and large deviation theorems for the activity vector. They give conditions for when the activity vector may be used to reconstruct the output pattern vector corresponding to the selected input pattern vector.

We consider the application of importance sampling in steady-state simulations of finite Markov chains. We show that, for a large class of performance measures, there is a choice of the alternative transition matrix for which the ratio of the variance of the importance sampling estimator to the variance of the naive simulation estimator converges to zero as the sample path length goes to infinity. Obtaining this ‘optimal’ transition matrix involves computing the performance measure of interest, so the optimal matrix cannot be computed in precisely those situations where simulation is required to estimate steady-state performance. However, our results show that alternative transition matrices of the form Q = P + E/T, where P is the original transition matrix and T is the sample path length, can be expected to provide good results. Moreover, we provide an iterative algorithm for obtaining alternative transition matrices of this form that converge to the optimal matrix as the number of iterations increases, and present an example that shows that spending some computer time iterating this algorithm and then conducting the simulation with the resulting alternative transition matrix may provide considerable variance reduction when compared to naive simulation.

In this paper, we give an alternative derivation of Kendall's representation for symmetric transition functions which relies on the backward and/or forward integral recursions. The proof uses a lemma concerning approximation by finite sections (which is useful in its own right) and is similar to the original proof for birth and death processes by Lederman and Reuter. Finally, we obtain a general result guaranteeing the existence of representations of transition functions such as those obtained by Pruitt and Iglehart.

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.

By making use of the geometric properties of simplex shape spaces, this paper investigates the problems relating to the estimation of the Fréchet means of the random shapes of simplices in Euclidean spaces and also, for the random shapes induced by certain normally distributed simplices, the problems relating to the location of these Fréchet means. In particular, we obtain an algorithm for computing sample mean shapes in simplex shape spaces which converges reasonably fast.

The starting point for this work was the following astonishing observation regarding stationary line processes in the plane.