# Approximating distributions and transient
probabilities of Markov chains by Bernstein
expolynomial functions

**András Horváth, Lorenzo Ridi, Enrico Vicario**

### Abstract:

In this extended abstract we consider the use of Bernstein polynomials
(BPs) for the approximation of distributions and transient probabilities of
continuous time Markov chains (CTMCs). We show that while standard BPs are
not appropriate to this purpose it is possible to derive from them a family
of functions, called in the sequel Bernstein expolynomials (BEs), which
enjoys those properties that are necessary for these approximations. For
what concerns distribution fitting, BEs correspond to a subset of the
family of matrix exponential distributions and hence they are of interest
in the field of matrix analytic methods. For what concerns transient
probabilities of CTMCs, BEs can provide closed-form approximations which
are useful in the analysis of models where the process subordinated to a
possibly non-Markovian period is described by a CTMC. The application of
BEs for approximating both distributions and transient probabilities will
be illustrated through several numerical examples.

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horvath
2010-11-29