74 research outputs found
Stochastic decomposition for -norm symmetric survival functions on the positive orthant
We derive a stochastic representation for the probability distribution on the
positive orthant whose association between components is minimal
among all probability laws with -norm symmetric survival functions. It
is given by a transformation of a uniform distribution on the standard unit
simplex that is multiplied with an independent finite mixture of certain beta
distributions and an additional atom at unity. On the one hand, this implies an
efficient simulation algorithm for arbitrary probability laws with
-norm symmetric survival function. On the other hand, this result is
leveraged to construct an exact simulation algorithm for max-infinitely
divisible probability distributions on the positive orthant whose exponent
measure has -norm symmetric survival function. Both applications
generalize existing results for the case to the case of arbitrary
Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes
We establish a correspondence between exchangeable sequences of random
variables whose finite-dimensional distributions are min- (or max-) infinitely
divisible and non-negative, non-decreasing, infinitely divisible stochastic
processes. The exponent measure of a min-id sequence is shown to be the sum of
a very simple "drift measure" and a mixture of product probability measures,
which corresponds uniquely to the L\'evy measure of a non-decreasing infinitely
divisible process. The latter is shown to be supported on non-negative and
non-decreasing functions. Our results provide an analytic umbrella which embeds
the de Finetti subfamilies of many classes of multivariate distributions, such
as exogenous shock models, exponential and geometric laws with lack-of-memory
property, min-stable multivariate exponential and extreme-value distributions,
as well as reciprocal Archimedean copulas with completely monotone generator
and Archimedean copulas with log-completely monotone generator.Comment: 53 pages, 3 Figure
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