As suggested in [107, p. 752], there is a connection between polynomials of binomial type and compound Poisson processes. Two different approaches can be found in [12,114].

A connection of polynomials of binomial type with renewal sequences can be found in [113].

Probabilistic aspects of Lagrange inversion and polynomials of binomial type can be found in [115].

Various probabilistic representations of Sheffer polynomials can be found in [22].

The following papers do not actually use umbral calculus, but provide interesting information on probabilistic aspect of generalized Appell polynomials [30,31,32,61].

sandro@win.tue.nl / loeb@labri.u-bordeaux.fr >

Non-parametric statistics (or distribution-free statistics) has a highly combinatorial flavor. In particular, lattice path counting techniques are often used. It is therefore not surprising that the main applications of umbral calculus to statistics are of a combinatorial nature [69,73,75,77].

Another application concerns statistics for parameters in power series distributions [15,56].

sandro@win.tue.nl / loeb@labri.u-bordeaux.fr