In this paper we study the Robbins–Monro procedure Xn+1 = Xn - an-1Yn with some fixed number a > 0 and establish the moderate deviation principle of the process {Xn}.

A strong approximation theorem is proved for some non-stationary complex-valued Gaussian processes and an explicit rate of convergence is achieved. The result answers a problem raised by S. Csörgő.

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.

The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.

In the excavation of archaeological remains of high value, a search tool with low cost can be used at first, but if we reach a stratum which is very likely to contain valuable material we must use a search tool with high efficiency to reduce the risk of destruction of the remains even if this is more expensive. Which tool should we use at each time? Furthermore, if the search costs too much in comparison with the reward of detection, it is not profitable to go on searching for ever. When should we stop the search? We answer these questions for continuous and discrete models.

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation process connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.

In this work we consider the mean-field traveling salesman problem, where the intercity distances are taken to be independent and identically distributed with some distribution F. We consider the simplest approximation algorithm, namely, the nearest-neighbor algorithm, where the rule is to move to the nearest nonvisited city. We show that the limiting behavior of the total length of the nearest-neighbor tour depends on the scaling properties of the density of F at 0 and derive the limits for all possible cases of general F.

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state (i0,j0), we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive a certain integral representation for the probability of this event, and an asymptotic expression for the case when i0 becomes large, a situation in which the event becomes highly unlikely. The integral representation follows from the solution of a boundary value problem and involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model, and the asymmetric exclusion process.

We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝn. Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

We calculate the mean throughput, number of collisions, successes, and idle slots for random tree algorithms with successive interference cancellation. Except for the case of the throughput for the binary tree, all the results are new. We furthermore disprove the claim that only the binary tree maximizes throughput. Our method works with many observables and can be used as a blueprint for further analysis.

The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction and a phase transition, and a lot can be learned about the process by studying its extinction time, $\tau_n$ , as a function of system size n. A number of existing results describe the scaling of $\tau_n$ as $n\to\infty$ for various choices of reproductive rate $r_n$ and initial population $X_n(0)$ as a function of n. We collect and complete this picture, obtaining a complete classification of all sequences $(r_n)$ and $(X_n(0))$ for which there exist rescaling parameters $(s_n)$ and $(t_n)$ such that $(\tau_n-t_n)/s_n$ converges in distribution as $n\to\infty$ , and identifying the limits in each case.

This paper is devoted to the study of regime-switching jump diffusion processes with countable regimes. It aims to establish Foster–Lyapunov-type criteria for exponential ergodicity of such processes. After recalling results concerning the petiteness of compact sets, this paper presents sufficient conditions for the existence of a Foster–Lyapunov function; this, in turn, helps to establish sufficient conditions for the desired exponential ergodicity for regime-switching jump diffusion processes. Finally, an application to feedback control problems is presented.

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear $\alpha$-stable Lévy processes under a sublinear expectation space with $\alpha\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry–Esseen bound and convergence rate for the $\alpha$-stable central limit theorem under sublinear expectation.

In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.

In this paper, we analyze a two-queue random time-limited Markov-modulated polling model. In the first part of the paper, we investigate the fluid version: fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switch-over time.

For this model, we first derive the Laplace–Stieltjes transform (LST) of the stationary marginal fluid content/workload at each queue. Subsequently, we derive a functional equation for the LST of the two-dimensional workload distribution that leads to a Riemann–Hilbert boundary value problem (BVP). After taking a heavy-traffic limit, and restricting ourselves to the symmetric case, the BVP simplifies and can be solved explicitly.

In the second part of the paper, allowing for more general (Lévy) input processes and server switching policies, we investigate the transient process limit of the joint workload in heavy traffic. Again solving a BVP, we determine the stationary distribution of the limiting process. We show that, in the symmetric case, this distribution coincides with our earlier solution of the BVP, implying that in this case the two limits (stationarity and heavy traffic) commute.

Aven and Jensen (1999) proposed an approximation to the downtime distribution for arbitrary coherent systems with binary and independent repairable components, where each component follows an alternating renewal process. The approximation is based on a mixture of the duration distributions of the cut sets in the system. We extend this approach to coherent systems composed by multistate independent components, where every component follows a semi-Markov process and where we introduce the concept of a minimal cut set in the multistate setting. We test our proposal on a simple power demand-generation system, by comparing the analytic approximation with simulation results, and we find it is accurate when the system is highly available.

Let ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$, where Y(t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.