Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws
Abstract:
The concept of a Lévy subordinator (non-decreasing paths, infinitely divisible (ID)
law at any point in time) is generalized to a family of non-decreasing stochastic processes
which are parameterized in terms of two Bernstein functions. Whereas the
independent increments property is only maintained in the Lévy subordinator special
case, the considered family is always strongly infinitely divisible with respect to
time (IDT), meaning that a path can be represented in distribution as a finite sum
with arbitrarily many summands of independent and identically distributed paths
of another process. Besides distributional properties of the process, we present two
applications to the design of accurate and efficient simulation algorithms, emphasizing
our interest in the investigated processes. First, each member of the considered
family corresponds uniquely to an exchangeable max-stable sequence of random
variables, and we demonstrate how the associated extreme-value copula can be simulated
exactly and effciently from its Pickands dependence measure. Second, we
show how one obtains different series and integral representations for infinitely divisible
probability laws by varying the parameterizing pair of Bernstein functions,
without changing the one-dimensional law of the process. As a particular example,
we present an exact simulation algorithm for compound Poisson distributions from
the Bondesson class, for which the generalized inverse of the distribution function
of the associated Stieltjes measure can be evaluated accurately.
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The concept of a Lévy subordinator (non-decreasing paths, infinitely divisible (ID)
law at any point in time) is generalized to a family of non-decreasing stochastic processes
which are parameterized in terms of two Bernstein functions. Whereas the
independent increments property is only maintained in the Lévy subordinator special
case, the considered family is always strongly infinitely divisible with respect to
time (IDT), meaning that a path can be represented in distribution as a finite...
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