We formulate and verify an almost-sure lattice renewal theorem for branching random walks, whose non-lattice analogue is originally due to Nerman. We also identify the limit in these renewal theorems (both lattice and non-lattice) as the limit of Kingman's well-known martingale multiplied by a deterministic factor.

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt:t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r < n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n3) (and, for practical purposes, even O(n2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted.

Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

Shaked (1982) gives a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports. However there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true. The difficulty arises because zero terms are discarded in the definition of the sign change operator S. Therefore the equality of and x′ does not imply that

With the aim of providing greater flexibility in developing and applying shot noise models, this paper studies shot noise on cluster point processes with both pointwise and cluster marks. For example, in financial modelling, responses to events in the financial market may occur in clusters, with random amplitudes including a ‘cluster component’ reflecting a degree of commonness among responses within a cluster. For such shot noise models, general formulae for the characteristic functional are developed and specialized to the case of Neyman-Scott clustering with cluster marks. For several general forms of response function, long range dependence of the corresponding equilibrium shot noise models is investigated. It is shown, for example, that long range dependence holds when the ‘structure component’ of the response function decays slowly enough, or when the response function has a finite random duration with a heavy tailed distribution.

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝd and a gauge body B ⊂ ℝd, such a generalized contact distribution is the conditional distribution of the random vector (dB(L,Z),uB(L,Z),pB(L,Z),lB(L,Z)) given that Z∩L = ∅, where Z is a Boolean model, dB(L,Z) is the distance of L from Z with respect to B, pB(L,Z) is the boundary point in L realizing this distance (if it exists uniquely), uB(L,Z) is the corresponding boundary point of B (if it exists uniquely) and lB(L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

We study the discrete-time approximation of doubly reflected backward stochastic differential equations (BSDEs) in a multidimensional setting. As in Ma and Zhang (2005) or Bouchard and Chassagneux (2008), we introduce the discretely reflected counterpart of these equations. We then provide representation formulae which allow us to obtain new regularity results. We also propose an Euler scheme type approximation and give new convergence results for both discretely and continuously reflected BSDEs.

This paper employs the methods currently used to solve many queuing control models in order to investigate the behavior of a firm's optimal advertising and pricing strategies over time. Given that a firm's market position expands or deteriorates in a probabilistic way which depends upon the current position, the rate of advertising, and the price the firm charges, we present conditions which ensure that the optimal level of advertising is a monotonic function of the firm's market position, and we discuss the economic meaning of these conditions. Furthermore, although the primary focus is upon a non-competitive environment, we develop the above model as a non-zero sum, two-person stochastic game and show that an equilibrium strategy exists which is simple to compute.

In this paper we treat a two-stage grouping procedure of building a k-out-of-n system from several clusters of components. We use a static framework in which the component reliabilities are fixed. Under such a framework, we address the impact of the selecting strategies, the sampling probabilities, and the component reliabilities on the constructed system’s reliability. An interesting finding is that the level of component reliabilities could be identified as a decisive factor in determining how the selecting strategies and the component reliabilities affect the system reliability. The new results generalize and extend those established earlier in the literature such as Di Crescenzo and Pellerey (2011), Hazra and Nanda (2014), Navarro, Pellerey, and Di Crescenzo (2015), and Hazra, Finkelstein, and Cha (2017). Several Monte Carlo simulation experiments are provided to illustrate the theoretical results.

In the biased annihilating branching process, particles place offspring on empty neighboring sites at rate A and destroy neighbors at rate 1. It is conjectured that for any λ ≥ 0 the population will spread to ∞, and this is shown in one dimension for The process on a finite graph when starting with a non-empty configuration has limiting distribution vλ /(λ +1), the product measure with density λ/(1 +λ). It is shown that vλ /(λ +1) and δ Ø are the only stationary distributions on Moreover, if and the initial configuration is non-empty, then the limiting measure is vλ /(λ +1) provided the initial measure converges.

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

Motivated by a wide range of applications, we consider a development of Whittle's restless bandit model in which project activation requires a state-dependent amount of a key resource, which is assumed to be available at a constant rate. As many projects may be activated at each decision epoch as resource availability allows. We seek a policy for project activation within resource constraints which minimises an aggregate cost rate for the system. Project indices derived from a Lagrangian relaxation of the original problem exist provided the structural requirement of indexability is met. Verification of this property and derivation of the related indices is greatly simplified when the solution of the Lagrangian relaxation has a state monotone structure for each constituent project. We demonstrate that this is indeed the case for a wide range of bidirectional projects in which the project state tends to move in a different direction when it is activated from that in which it moves when passive. This is natural in many application domains in which activation of a project ameliorates its condition, which otherwise tends to deteriorate or deplete. In some cases the state monotonicity required is related to the structure of state transitions, while in others it is also related to the nature of costs. Two numerical studies demonstrate the value of the ideas for the construction of policies for dynamic resource allocation, most especially in contexts which involve a large number of projects.

We show that the stationary version of the queueing relation H = λG is equivalent to the basic Palm transformation for stationary marked point processes.

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L−⊂L, where L is an open set in ℂd, and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

Rough volatility is a well-established statistical stylized fact of financial assets. This property has led to the design and analysis of various new rough stochastic volatility models. However, most of these developments have been carried out in the mono-asset case. In this work, we show that some specific multivariate rough volatility models arise naturally from microstructural properties of the joint dynamics of asset prices. To do so, we use Hawkes processes to build microscopic models that accurately reproduce high-frequency cross-asset interactions and investigate their long-term scaling limits. We emphasize the relevance of our approach by providing insights on the role of microscopic features such as momentum and mean-reversion in the multidimensional price formation process. In particular, we recover classical properties of high-dimensional stock correlation matrices.

The nonstationary Erlang-A queue is a fundamental queueing model that is used to describe the dynamic behavior of large-scale multiserver service systems that may experience customer abandonments, such as call centers, hospitals, and urban mobility systems. In this paper we develop novel approximations to all of its transient and steady state moments, the moment generating function, and the cumulant generating function. We also provide precise bounds for the difference of our approximations and the true model. More importantly, we show that our approximations have explicit stochastic representations as shifted Poisson random variables. Moreover, we are also able to show that our approximations and bounds also hold for nonstationary Erlang-B and Erlang-C queueing models under certain stability conditions.