We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.

We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

We introduce and study the notion of sure profits via flash strategies, consisting of a high-frequency limit of buy-and-hold trading strategies. In a fully general setting, without imposing any semimartingale restriction, we prove that there are no sure profits via flash strategies if and only if asset prices do not exhibit predictable jumps. This result relies on the general theory of processes and provides the most general formulation of the well-known fact that, in an arbitrage-free financial market, asset prices (including dividends) should not exhibit jumps of a predictable direction or magnitude at predictable times. We furthermore show that any price process is always right-continuous in the absence of sure profits. Our results are robust under small transaction costs and imply that, under minimal assumptions, price changes occurring at scheduled dates should only be due to unanticipated information releases.

This paper deals with the first passage optimality and variance minimisation problems of discrete-time Markov decision processes (MDPs) with varying discount factors and unbounded rewards/costs. First, under suitable conditions slightly weaker than those in the previous literature on the standard (infinite horizon) discounted MDPs, we establish the existence and characterisation of the first passage expected-optimal stationary policies. Second, to further distinguish the expected-optimal stationary policies, we introduce the variance minimisation problem, prove that it is equivalent to a new first passage optimality problem of MDPs, and, thus, show the existence of a variance-optimal policy that minimises the variance over the set of all first passage expected-optimal stationary policies. Finally, we use a computable example to illustrate our main results and also to show the difference between the first passage optimality here and the standard discount optimality of MDPs in the previous literature.

Simple random walks on a partially directed version of Z2 are considered. More precisely, vertical edges between neighbouring vertices of Z2 can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function; the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.

Different Markov chains can be used for approximate sampling of a distribution given by an unnormalized density function with respect to the Lebesgue measure. The hit-and-run, (hybrid) slice sampler, and random walk Metropolis algorithm are popular tools to simulate such Markov chains. We develop a general approach to compare the efficiency of these sampling procedures by the use of a partial ordering of their Markov operators, the covariance ordering. In particular, we show that the hit-and-run and the simple slice sampler are more efficient than a hybrid slice sampler based on hit-and-run, which, itself, is more efficient than a (lazy) random walk Metropolis algorithm.

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

In a two-person red-and-black game, each player holds an integral amount of chips. At each stage of the game, each player can bet any integral amount in his possession, winning the chips of his opponent with a probability which is a function of the ratio of his bet to the sum of both players' bets and is called a win probability function. Both players seek to maximize the probability of winning the entire fortune of his opponent. In this paper we propose two new models. In the first model, at each stage, there is a positive probability that two players exchange their bets. In the second model, the win probability functions are stage dependent. In both models, we obtain suitable conditions on the win probability functions such that it is a Nash equilibrium for the subfair player to play boldly and for the superfair player to play timidly.

Correlated random walks provide an elementary model for processes that exhibit directional reinforcement behavior. This paper develops optimal multiple stopping strategies - buy/sell rules - for correlated random walks. The work extends previous results given in Allaart and Monticino (2001) by considering random step sizes and allowing possibly negative reinforcement of the walk's current direction. The optimal strategies fall into two general classes - cases where conservative buy-and-hold type strategies are optimal and cases for which it is optimal to follow aggressive trading strategies of successively buying and selling the commodity depending on whether the price goes up or down. Simulation examples are given based on a stock index fund to illustrate the variation in return possible using the theoretically optimal stop rules compared to simpler buy-and-hold strategies.

Birth and death processes, i.e. Markov chains {X(t)}, 0 ≦ t ≦ ∞, taking values in the set I of non-negative integers, for which the infinitesimal transition probabilities are have found wide application in mathematical ecology . In order that the infinite system of differential Equations (1) should have solutions such that Pij(t) ≦ 0, all t and it is necessary and sufficient that . The conditions ensure that with probability one, the sample paths of X(t) have no discontinuities worse than jumps, and in this case the solution is unique. If (3) does not hold then for some t1 at least, there is a set of sample paths of positive probability, each of which has t1 as a limit point of jumps. For pure birth processes, (in which μj = 0) this clearly implies X(t) → ∞ as t → t1.

If X1, …, Xn obey a multivariate normal distribution with zero means, then the probability that Xi > a for all i = 1, …, n is often called a multivariate normal integral. Such integrals have been considered by various investigators, particularly when a = 0. (For a bibliography, see Gupta.)

It is well known that each statistic in the family of power divergence statistics, across n trials and r classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio, and Freeman–Tukey statistics correspond to $\lambda=1,0,-1/2$ , respectively), is asymptotically chi-square distributed as the sample size tends to infinity. We obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample n, and all index parameters ( $\lambda>-1$ ) for which such finite-sample bounds are meaningful. We obtain bounds that are of the optimal order $n^{-1}$ . The dependence of our bounds on the index parameter $\lambda$ and the cell classification probabilities is also optimal, and the dependence on the number of cells is also respectable. Our bounds generalise, complement, and improve on recent results from the literature.

We study convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions. We give a method for reducing the convergence to the tail behavior of the lifetime via a property we call the first hitting uniqueness. We apply the results to Kummer diffusions with negative drift and give a class of initial distributions converging to each non-minimal quasi-stationary distribution.

We investigate the asymptotic behaviour of homogeneous multidimensional Markov chains whose states have nonnegative integer components. We obtain growth rates for these models in a situation similar to the near-critical case for branching processes, provided that they converge to infinity with positive probability. Finally, the general theoretical results are applied to a class of controlled multitype branching process in which random control is allowed.