Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

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Combinatorial Stochastic Processes

In this article, we present a concise review of developments on various continuous multivariate distributions. We first present some basic definitions and notations. Then, we present several important continuous multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
conditional distribution;
truncated distribution;
regression;
bivariate normal;
multivariate normal;
multivariate exponential;
multivariate gamma;
dirichlet;
inverted dirichlet;
liouville;
multivariate logistic;
multivariate pareto;
multivariate extreme value;
multivariate t;
wishart translated systems;
multivariate exponential families

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Continuous Multivariate Distributions

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

A distribution with rational Laplace-Stieltjes transform is of phase type if and only if it is either the point mass at zero, or it has a continuous positive density on the positive reals and its Laplace-Stieltjes transform has a unique pole of maximal real part (which is therefore real). This result is proved, and the corresponding characterization of discrete phase-type distributions is stated and proved. Our methods are based on a geometric property of the set of phase-type distributions associated with a Markov chain.

Characterization of phase-type distributions

A phase-type distribution is the distribution of a killing time in a finite-state Markov chain. This paper describes some conjectures concerning these distributions. Some partial proofs and other evidence are offered in support of the conjectures. Some of the conjectures concern the geometry of the set of phase-type distributions, while others concern the properties of their densities

Phase-type distributions: open problems and a few properties

We introduce two properties which are useful in addressing the question of non-uniqueness of representations of phase-type distributions. One property, called phase-type simplicity, concerns the possibility that a given phase-type distribution may have two distinct representations in terms of the same Markov chain. The second, called phase-type majorization, concerns the possibility that one Markov chain may provide representations for all the distributions represented by another Markov chain. We prove some characterizations of these properties, discuss their interrelationship, and relate them to some known results on mixtures of convolutions of exponential distributions.

On non-uniqueness of representations of phase-type distributions

The M/M/oo queue in a random environment is an infinite-server queue where arrival and service rates are stochastic processes. Here we study the steady-state behavior of such a system. Explicit results are obtained for the factorial moments, the impossibility of a 'matrix-Poisson' steady-state distribution is demonstrated and two numerical examples are presented.

The M/M /∞ queue in a random environment

Grassmann, Taksar, and Heyman introduced a variant of Gaussian climination for computing the steady-state vector of a Markov chain. In this paper we prove that their algorithm is stable, and that the problem itself is well-conditioned, in the sense of entrywise relative error. Thus the algorithm computes each entry of the steady-state vector with low relative error. Even the small steady-state probabilities are computed accurately. The key to our analysis is to focus on entrywise relative error in both the data and the computed solution, rather than making the standard assessments of error based on norms. Our conclusions do not depend on any Condition numbers for the problem.

Colm Art O'Cinneide

Papers

Characterization of phase-type distributions

Phase-type distributions: open problems and a few properties

On non-uniqueness of representations of phase-type distributions

The M/M /∞ queue in a random environment

Entrywise perturbation theory and error analysis for Markov chains