We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

If X is a Brownian motion with drift and γ = inf{t > 0: Mt = t} we derive the joint density of the triple {U, γ, Δ}, where and Δ= γ —Xγ. In the case δ = 0 it follows easily from this that Δ has an Exp(2) distribution and this in turn implies the rather surprising result that if τ= inf{t > 0: Xt = Mt = t}, then Pr{τ = 0} = 0 and . We also derive various other distributional results involving the pair (X, M), including for example the distribution of ; in particular we show that, in case δ. = 1, when Pr{0 < τ < ∞} = 1, the ratio τ+/τ has the arc-sine distribution.

The concept of isotropically random mosaic introduced by Pielou, is recalled. The notion of anisotropically random mosaic with respect to a point is defined. An example is given.

This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ESτ = ES1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

We consider a family of M(t)/M(t)/1/1 loss systems with arrival and service intensities (λt (c), μt (c)) = (λct, μct), where (λt, μt) are governed by an irreducible Markov process with infinitesimal generator Q = (qij)m × m such that (λt, μt) = (λi, μi) when the Markov process is in state i. Based on matrix analysis we show that the blocking probability is decreasing in c in the interval [0, c∗], where c∗ = 1/maxi Σj≠iqij/(λi + μi). Two special cases are studied for which the result can be extended to all c. These results support Ross's conjecture that a more regular arrival (and service) process leads to a smaller blocking probability.

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

This study is motivated by problems of molecular sequence comparison for multiple marker arrays with correlated distributions. In this paper, the model assumes two (or more) kinds of markers, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say ≥ m) occur, and (ii) few of Marker B (say ≤ l) occur. We title these the conditional r-scan models, and inquire on the extent to which Marker A clusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremal r-scan statistics from the A array satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.

Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

An algorithm is suggested for approximating the performance of a D/G/K loss system with deterministic input, generally distributed processing time, K heterogeneous servers, the random access processing discipline, and retrials in steady state. In loss systems with retrials, the units which at the instants of their arrival at the system find all the servers busy, are not lost: those units retry to be processed by merging with the incoming arrival units. In this system, a fraction of the units which have not initially been processed will be allowed to leave the system. The performance of this system in steady state is approximated by a recursive technique.

The properties of frequency distributions of inter-event intervals in the pooled output of several point processes where the points can be distinguished are investigated. In the case where the component processes are mutually independent, the p.d.f.'s of inter-event intervals in the pooled output are explicitly expressed, and the reciprocal relations on the p.d.f.'s and on the numbers of inter-event intervals are revealed.

De Bruijn's identity relates two important concepts in information theory: Fisher information and differential entropy. Unlike the common practice in the literature, in this paper we consider general additive non-Gaussian noise channels where more realistically, the input signal and additive noise are not independently distributed. It is shown that, for general dependent signal and noise, the first derivative of the differential entropy is directly related to the conditional mean estimate of the input. Then, by using Gaussian and Farlie–Gumbel–Morgenstern copulas, special versions of the result are given in the respective case of additive normally distributed noise. The previous result on independent Gaussian noise channels is included as a special case. Illustrative examples are also provided.

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).

The jigsaw percolation process on graphs was introduced by Brummitt et al. (2015) as a model of collaborative solutions of puzzles in social networks. Percolation in this process may be viewed as the joint connectedness of two graphs on a common vertex set. Our aim is to extend a result of Bollobás et al. (2017) concerning this process to hypergraphs for a variety of possible definitions of connectedness. In particular, we determine the asymptotic order of the critical threshold probability for percolation when both hypergraphs are chosen binomially at random.

Feller (1968) showed that the probability functions of each of three random variables associated with coin-tossing (independent, simple, symmetric random walk) are (a) the same (except at the end points), (b) approximately arc-sine. In this paper these two properties are shown to hold for Markov-dependent coin-tossing (Markov-dependent, simple, symmetric random walk).

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

The paper studies a network of queues in which units arrive singly in a Poisson stream at a service channel S from which they branch out into k parallel channels S1, S2, …, Sk. After having been serviced at S1, S2, … Sk, the units converge again into a single channel S′. The service times of units at each of the channels are assumed to be exponential. Units finally serviced at S’ may leave the system or may again join S. This has been considered by taking two models denoted as Model A and Model B. Steady-state probabilities giving the number of units present in the system have been obtained explicitly for both the models. The expressions for mean queue lengths have also been arrived at.

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.