In the last ten years, the employment of symbolic methods has substantially
extended both the theory and the applications of statistics and probability.
This survey reviews the development of a symbolic technique arising from
classical umbral calculus, as introduced by Rota and Taylor in 1994. The
usefulness of this symbolic technique is twofold. The first is to show how new
algebraic identities drive in discovering insights among topics apparently very
far from each other and related to probability and statistics. One of the main
tools is a formal generalization of the convolution of identical probability
distributions, which allows us to employ compound Poisson random variables in
various topics that are only somewhat interrelated. Having got a different and
deeper viewpoint, the second goal is to show how to set up algorithmic
processes performing efficiently algebraic calculations. In particular, the
challenge of finding these symbolic procedures should lead to a new method, and
it poses new problems involving both computational and conceptual issues.
Evidence of efficiency in applying this symbolic method will be shown within
statistical inference, parameter estimation, L\'evy processes, and, more
generally, problems involving multivariate functions. The symbolic
representation of Sheffer polynomial sequences allows us to carry out a
unifying theory of classical, Boolean and free cumulants. Recent connections
within random matrices have extended the applications of the symbolic method.Comment: 72 page
