A class of infinitely divisible variance functions with an application to the polynomial case

Let be a natural exponential family on ??? with variance function (V, Ω). Here, Ω is the mean domain of and V is its variance expressed in terms of the mean μ ε Ω. In this note we prove the following result. Consider an open interval Ω = (0, b), 0 < b ∞, and a positive real analytic function V on Ω. If V² is absolutely monotone on [0, b) and V has the form μt(μ), where 1 and t is real analytic in a neighborhood of zero, then there exits an infinitely divisible natural exponential family with variance function (V, Ω). We illustrate this result with several examples of general nature.     

Generalization to the sufficient conditions for a discrete random variable to be infinitely divisible

The convenient sufficient conditions for the infinite divisibility of discrete distributions are generalized. A numerical example is presented. The role of infinitely divisible distributions in applications is discussed.     

Asymptotics of the density of an infinitely divisible distribution at infinity

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, Trudy Seminara, pp. 123–131, 1990.     

On a certain class of infinitely divisible distributions

We characterize the class of distribution functions Φ(x), which are limits in the following sense: there exist a sequence of independent and equally distributed random variables {ξ _n }, numerical sequences {a _k }, {b _k } and natural numbers {n _k } such that $$\mathop {lim}\limits_{k \to \infty } Prob\left\{ {\frac{1}{{a_k }}\mathop {\Sigma }\limits_{k = 1}^{n_k } \xi _k - b_k< x} \right\} = \Phi (x)$$ and $$\mathop {\lim \inf }\limits_{k \to \infty } (n_k /n_{k + 1} ) > 0$$ .     

Testing independence of variates in an infinitely divisible random vector

A method is given for testing the independence of variates in an infinitely divisible random vector and for testing the independence of several subsets of the variates. Applications to stochastic processes are indicated.     

On an application of infinitely divisible distributions to quadrature problems

— - , .     

A generalisation of variable upper bounding and generalised upper bounding

Constraints of the form Σx_i?y where the variable y may appear in any number of constraints may be considered to be generalisations both of Generalised Upper Bounds (GUB) and Variable Upper Bounds (VUB). A method of representing such constraints implicity in linear programs is demonstrated. The possibility of the implementation of such constraints into an advanced linear programming system is considered and shown to present no major difficulties.     

A condition for absolute continuity of infinitely divisible distribution functions

Summary Throughout this paper the symbols r.v., d.f., ch.f., and i.d. will stand, respectively, for random variable, distribution function, characteristic function, and infinitely divisible.Let F(x) be an i.d.d.f. Hartman and Wintner [5] and Blum and Rosenblatt [1] have given a condition, necessary and sufficient, for F(x) to be a continuous d.f. In this note a sufficient condition for F(x) to be an absolutely continuous d.f. is given.Research supported by ONR Contract No. NONR-285(46).Research supported in part by a National Science Foundation fellowship.     

A weighted version of a rearrangement inequality

We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then Dirichlet-type integrals of sufficiently smooth real-valued functions decrease under an appropriate equimeasurable rearrangement.

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Sunto Si dimostra che se una misura, positiva e assolutamente continua, rende valida una certa disuguaglianza isoperimetrica relativa, allora integral del tipo di Dirichlet di funzioni sufficientemente regolari descrescono per effetto di un opportuno riordinamento equidistribuito.

     

On the structure of regular infinitely divisible point processes

A representation for the probability generating functional (p.g.fl.) of a regular infinitely divisible (i.d.) stochastic point process, motivated as a generalization of the Gauss-Poisson process, is presented. The functional is characterized by a sequence of Borel product measures. Necessary and sufficient conditions, in terms of these Borel measures, are given for this representation to be a p.g.fl., thus characterizing all regular i.d. point processes.     

Characterization of a class of infinitely divisible distributions in Hilbert space

It is established that the spectral measure of an infinitely divisible distribution F in a Hilbert space H is concentrated in a sphere of finite radius if and only if the integral ∫ _H exp (α∥x∥ In (∥x∥+1))dF is finite for some numberα>0. If this integral is finite for anyα>0 then the infinitely divisible distribution F is normal (maybe, degenerate).     

On a characterization of infinitely divisible characteristic functionals on a Hubert space

Summary A characterization of infinitely divisible characteristic functional on a Hilbert space, analogous to that of Johansen [1], is given.     

A lenticular version of a von Neumann inequality

We generalize to lens-shaped domains the classical von Neumann inequality for the disk. Received: 29 March 2005; revised: 14 June 2005