Let 𝓅 be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that Gn, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With Gn, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

We present two variations of a two-person, noncooperative stochastic game, inspired by the famous red-and-black gambling problem presented by Dubins and Savage. Two players each hold an integer amount of money and they each aim to win the other player's fortune. At every stage of the game they simultaneously bid an integer portion of their current fortune, and their probabilities of winning depend on these bids. We describe two different laws of motion specifying this dependency. In one version of the game, the players' probabilities of winning are proportional to their bets. In the other version, the probabilities of winning depend on the size of their bets and a weight parameter w. For each version we give a Nash equilibrium, in which the player for which the game is subfair (w ≤ ½) plays boldly and the player for which the game is superfair (w ≥ ½) plays timidly.

A simplified proof of Thorp and Walden's fundamental theorem of card counting is presented, and a corresponding central limit theorem is established. Results are applied to the casino game of trente et quarante, which was studied by Poisson and De Morgan.

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

We prove the existence and uniqueness of the solution to certain reflected backward stochastic differential equations (RBSDEs) with one continuous barrier and deterministic terminal time, under monotonicity, and general increasing growth conditions on the associated coefficient. As an application, we obtain, in some constraint cases, the price of an American contingent claim as the unique solution of such an RBSDE.

In this paper, we analyze the diffusion limit of a discrete-time queueing system with constant service rate and connections that randomly enter and depart from the system. Each connection generates periodic traffic while it is active, and a connection's lifetime has finite mean. This can model a time division multiple access system with constant bit-rate connections. The diffusion scaling retains semiperiodic behavior in the limit, allowing for both short-time analysis (within one frame) and long-time analysis (over multiple frames). Weak convergence of the cumulative arrival process and the stationary buffer-length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discrete-time stationary-increment Gaussian process interpolated by Brownian bridges. We present bounds on the overflow probability of the limit queueing process as functions of the arrival rate and the connection lifetime distribution. Also, numerical and simulation results are presented for geometrically distributed connection lifetimes.

Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R+n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

Let Yk(ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0Yk(ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Yk. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Yk(ω) := ∑j=0kYj(ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Ykk, and show that, as n → ∞, Yk/nk is asymptotically Gaussian under the conditional distribution P(· | Z = n).

The first purpose of this article is to study the phase transitions of a new interacting particle system. We consider two types of particle, each of which gives birth to particles of the same type as the parent. Particles of the second type can die, whereas those of the first type mutate into the second type. We prove that the three possible outcomes of the process, that is, extinction, survival of the type-2s, or coexistence, may each occur, depending on the selected parameters. Our second, and main, objective, however, is to investigate the duality properties of the process; the corresponding dual process exhibits a structure somewhat different to that of well-known particle systems.