Infinitely divisible measures onp-adic groups

We show that any infinitely divisible measure on ap-adic algebraic group (p a prime) has a translatex, by an elementx centralizing the support of , such thatx can be embedded in a continuous real one-parameter semigroup {V _t } _t >0, asx=v ₁.     

Infinitely divisible stochastic processes

Summary It often happens that a stochastic process may be approximated by a sum of a large number of independent components no one of which contributes a significant proportion of the whole. For example the depth of water in a lake with many small rivers flowing into it from distant sources, or the point process of calls entering a telephone exchange (considered as the sum of a number of point processes of calls made by individual subscribers) may approximately fulfil these hypotheses. In the present work we formulate and solve the problem of characterizing stochastic processes all of whose finite-dimensional distributions are infinitely divisible. Although some of our results could be derived from known theorems on probabilities on general algebraic structures, many could not and it seems preferable to take the vector-valued infinitely divisible laws as our starting point. We emphasize that an infinitely divisible process (in our sense) on the real line is not necessarily a decomposable process in the sense of Lévy (cf. § 4) though decomposable processes are particular cases.In § 1 a representation theorem for non-negative (and possibly infinite) stochastic processes is derived, while an analogous theorem in the real-valued case is to be found in § 2. § 3 is devoted to the statement of a central limit theorem and the investigation of some of the properties of infinitely divisible processes. The investigation is continued in § 4 by an examination of processes on the real line giving, for example, a representation theorem under weak conditions for infinitely divisible processes which are a.s. sample continuous. Finally in § 5 a study is made of infinitely divisible point processes and random measures.The author is indebted to Professor J. F. C. Kingman for advice and encouragement.     

Infinitely divisible completely positive mappings

Summary We generalise the theory of infinitely divisible positive definite functions f:G on a group G to a theory of infinite divisibility for completely positive mappings : G() taking values in the algebra of bounded operators on some Hilbert space .We prove a structure theorem for normalised infinitely divisible completely positive mappings which shows that the mapping , its Stinespring representation and its Stinespring isometry are of type S (in the sense of Guichardet [Gui]). Furthermore, we prove that a completely positive mapping is infinitely divisible if and only if it is the exponential (as defined in this paper) of a hermitian conditionally completely positive mapping.     

Infinitely divisible and stable statistical experiments

The purpose of the present contribution to the theory of comparison of statistical experiments is twofold: to describe in a somewhat direct way the functional-analytic approach to the central limit theorem for experiments in terms of infinitesimal triangular arrays, and at the same time to emphasize the application of an indisputable definition of Poisson experiments. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Infinitely divisible probabilities on linearp-adic groups

In this paper we extend the work of Shah, on the structure of infinitely divisible probabilities onp-adic linear groups, to give a classification for all such probabilities.     

Positive vector measures with given marginals

Suppose E is an ordered locally convex space, X ₁ and X ₂ Hausdorff completely regular spaces and Q a uniformly bounded, convex and closed subset of M _t ⁺ (X ₁ × X ₂, E). For i = 1, 2, let μ _i   M _t ⁺ (X _i , E). Then, under some topological and order conditions on E, necessary and sufficient conditions are established for the existence of an element in Q, having marginals μ ₁ and μ ₂.     

Infinitely divisible probabilities on discrete linear groups

We investigate the structure of infinitely divisible probability measures on a discrete linear group. It is shown that for any such measure there is an infinitely divisible elementz in the centralizer of the support of the measure, such that the translate of the measure byz is embeddable over the subgroup generated by the support of the measure. Examples are given to show that this reult is best possible.     

Infinitely divisible distributions over locally compact non-archimedean fields

This paper is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite, locally compact fields of zero and positive characteristics with nontrivial non-Archimedean norms. Infinitely divisible distributions are studied. Theorems about their characteristic functionals are proved. Particular cases are demonstrated as applications to non-Archimedean analogs of Gaussian and Poisson processes and their generalizations.     

Extremal measures with prescribed marginals (finite case)

The problem of describing the extremal mesures with given marginals on a finite Cartesian product is studied.     

Worst VaR scenarios with given marginals and measures of association

This paper studies the problem of finding best-possible upper bounds on the Value-at-Risk for a function of two random variables when the marginal distributions are known and additional nonparametric information on the dependence structure, such as the value of a measure of association, is available. The same problem for the Tail-Value-at-Risk is also briefly discussed.