In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

In earlier work, we investigated the dynamics of shape when rectangles are split into two. Further exploration, into the more general issues of Markovian sequences of rectangular shapes, has identified four particularly appealing problems. These problems, which lead to interesting invariant distributions on [0,1], have motivating links with the classical works of Blaschke, Crofton, D. G. Kendall, Rényi and Sulanke.

We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.

We consider the tail probabilities of a class of compound distributions. First, the relations between reliability distribution classes and heavy-tailed distributions are discussed. These relations reveal that many previous results on estimating the tail probabilities are not applicable to heavy-tailed distributions.

Then, a generalized Wald's identity and identities for compound geometric distributions are presented in terms of renewal processes. Using these identities, lower and upper bounds for the tail probabilities are derived in a unified way for the class of compound distributions, both under the conditions of NBU and NWU tails, which include exponential tails, as well as under the condition of heavy-tailed distributions.

Finally, simplified bounds are derived by the technique of stochastic ordering. This method removes some unnecessary technical assumptions and corrects errors in the proof of some previous results.

In this paper we describe a model for survival functions. Under this model a system is subject to shocks governed by a Poisson process. Each shock to the system causes a random damage that grows in time. Damages accumulate additively and the system fails if the total damage exceeds a certain capacity or threshold. Various properties of this model are obtained. Sufficient conditions are derived for the failure rate (FR) order and the stochastic order to hold between the random lifetimes of two systems whose failures can be described by our proposed model.

The classical martingale characterizations of the Poisson process were obtained for point process or purely discontinuous martingale i.e. under additional assumptions on properties of trajectories. Here our aim is to search for related characterizations without relying on properties of trajectories. Except for a new martingale characterization, results based on conditional moments jointly involving the past and the nearest future are presented.

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. L p -distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

We will state a general version of Simpson's paradox, which corresponds to the loss of some dependence properties under marginalization. We will then provide conditions under which the paradox is avoided. Finally we will relate these Simpson-type paradoxes to some well-known paradoxes concerning the loss of ageing properties when the level of information changes.

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

The study of the distribution of the distance between words in a random sequence of letters is interesting in view of application in genome sequence analysis. In this paper we give the exact distribution probability and cumulative distribution function of the distances between two successive occurrences of a given word and between the nth and the (n+m)th occurrences under three models of generation of the letters: i.i.d. with the same probability for each letter, i.i.d. with different probabilities and Markov process. The generating function and the first two moments are also given. The point of studying the distances instead of the counting process is that we get some knowledge not only about the frequency of a word but also about its longitudinal distribution in the sequence.

Interest has been shown in Markovian sequences of geometric shapes. Mostly the equations for invariant probability measures over shape space are extremely complicated and multidimensional. This paper deals with rectangles which have a simple one-dimensional shape descriptor. We explore the invariant distributions of shape under a variety of randomised rules for splitting the rectangle into two sub-rectangles, with numerous methods for selecting the next shape in sequence. Many explicit results emerge. These help to fill a vacant niche in shape theory, whilst contributing at the same time, new distributions on [0,1] and interesting examples of Markov processes or, in the language of another discipline, of stochastic dynamical systems.

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalizing puzzle in using Markov chain knowledge to view the celebrated Stein–Chen method for Poisson approximations. This work complements the work of Barbour (1988) for the case of Poisson random variable approximation.

Let ζ be a Markov chain on a finite state space D, f a function from D to ℝd , and S n = ∑k=1 n f(ζk ). We prove an invariance theorem for S and derive an explicit expression of the limit covariance matrix. We give its exact value for p-reinforced random walks on ℤ2 with p = 1, 2, 3.

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, αn of E. Choose n independent points β1, β2, …, βn randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, αn ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

Let pα,θ be the Linnik density, that is, the probability density with the characteristic function . The following problem is studied: Let (α θ), (β, ϑ) be two point of PD. When is it possible to represent β,ϑ as a scale mixture of pαθ? A subset of the admissible pairs (α, θ), (β, ϑ) is described.

This paper deals with existence of bivariate Fréchet optimal lower bounds for two sets of events, and provides a practical approach to find this kind of bound. The main device used is linear programming ideas, coupled with construction of probability spaces. The highlight of this paper is that perturbation terms in the optimization process, even when a tie occurs, are not necessary in this practical implementation.

Let X = (X 1, …, X n ) be a random binary vector, with a known joint distribution P. It is necessary to inspect the coordinates sequentially in order to determine if X i = 0 for every i, i = 1, …, n. We find bounds for the ratio of the expected number of coordinates inspected using optimal and greedy searching policies.

The inverse absorption distribution is shown to be a q-Pascal analogue of the Kemp and Kemp (1991) q-binomial distribution. The probabilities for the direct absorption distribution are obtained via the inverse absorption probabilities and exact expressions for its first two factorial moments are derived using q-series transformations of its probability generating function. Alternative models for the distribution are given.