Generalized Gamma Convolutions and Related Classes of Distributions and Densities

Bondesson, Lennart

doi:10.1007/978-1-4612-2948-3

Cited by 252 publications

(423 citation statements)

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“…The following is a minor generalization of the observation in Bondesson (1992), p. 19. For future references we state it as follows: PROPOSITION 1.…”

Section: Selfdecomposability and The Strong Markov Propertymentioning

confidence: 54%

Remarks on the Selfdecomposability and New Examples

Jurek¹

2001

Demonstratio Mathematica

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Abstract. The analytic property of the seljdecomposability of characteristic functions is presented from stochastic processes point of view. This provides new examples or proofs, as well as a link between the stochastic analysis and the theory of characteristic functions. A new interpretation of the famous Levy's stochastic area formula is given. Introduction and notationsThe class of selfdecomposable probability distributions, denoted as SD, (known also as the class L or Levy class L distributions), appears in the theory of limiting distributions as laws of normalized partial sums of independent random variables but not necessarily identically distributed. However, the additional assumption of the infinitesimality of the summands guarantees their infinite divisibility; cf. Jurek & Mason (1993), Section 3.3.9.All our random variables or stochastic processes are defined on a fixed probability space (fI, T, V). For a given random variable X (for short: rv) or its probability distribution /x = C(X) or its probability density /, provided it exits (i.e.,d/j,(x) = f(x)dx), we define its characteristic function (in short: char.f.) 4>x{t) = 4>(t) as follows dV(cj) = J e itx dfi(x), t € R.n R Research supported in part by Grant no. 2 P03 A02914 from KBN, Warsaw, Poland.

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“…The following is a minor generalization of the observation in Bondesson (1992), p. 19. For future references we state it as follows: PROPOSITION 1.…”

Section: Selfdecomposability and The Strong Markov Propertymentioning

confidence: 54%

Remarks on the Selfdecomposability and New Examples

Jurek¹

2001

Demonstratio Mathematica

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“…This is discussed in Feller (1971, VII.6) and plays a prominent role in the analysis of infinite-divisibility by Thorin (1977Thorin ( ,1978 and Bondesson (1979Bondesson ( , 1992. We will discuss some features of this in the forthcoming sections.…”

Section: Posterior Distributionsmentioning

confidence: 91%

Bayesian Inference via Classes of Normalized Random Measures

2005

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One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by an appropriate normalization. Specifically we achieve explicit and tractable forms of the posterior and the marginal distributions, including an explicit and easily used description of generalizations of the important Blackwell-MacQueen Pólya urn distribution. Such simplifications are achieved by the use of a latent variable which admits quite interesting interpretations which allow to gain a better understanding of the behaviour of these random probability measures. It is noteworthy that these models are generalizations of models considered by Kingman (1975) in a non-Bayesian context. Such models are known to play a significant role in a variety of applications including genetics, physics, and work involving random mappings and assemblies. Hence our analysis is of utility in those contexts as well. We also show how our results may be applied to Bayesian mixture models and describe computational schemes which are generalizations of known efficient methods for the case of the Dirichlet process. We illustrate new examples of processes which can play the role of priors for Bayesian nonparametric inference and finally point out some interesting connections with the theory of generalized gamma convolutions initiated by Thorin and further developed by Bondesson.

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“…A sufficient condition is given by the following theorem. This result uses the Thorin class of infinitely divisible distributions, also known as the class of generalized gamma convolutions, which has been systematically studied by Bondesson (1992 …”

Section: Proofmentioning

confidence: 99%