# Theory of Stochastic Objects Probability, Stochastic Processes and Inference by Athanasios Christou Micheas

Random variables and random vectors have been well defined and studied for
over a century. Subsequently, in the history of statistical science, researchers began
considering collections of points together, which gave birth to point process theory
and more recently, to random set theory. This was mainly motivated due to advances
in technology and the types of data that experimenters began investigating, which
in turn led to the creation and investigation of advanced statistical methods able to
handle such data.
In this book we take the reader on a journey through some of the most essential
topics in mathematics and statistics, constantly building on previous concepts, mak-
ing the transition from elementary statistical inference to the advanced probabilistic
treatment more natural and concrete. Our central focus is defining and exploring the
concept of a random quantity or object in different contexts, where depending on
the data under consideration, “random objects” are described using random vari-
ables, vectors or matrices, stochastic processes, integrals and differential equations,
or point processes and random sets.
This view of random objects has not been adequately investigated and pre-
sented in mathematics and statistics textbooks that are out there since they have
mostly concentrated on specific parts of the aforementioned concepts. This is one
of the reasons why I undertake the task of writing a textbook that would present
the knowledge in a concrete way, through examples and exercises, which is sorely
needed in understanding statistical inference, probability theory and stochastic pro-
cesses. This approach will help the instructor of these topics to engage the students
through problem sets and present the theory and applications involved in a way that
they will appreciate.
Since this monumental task cannot be accomplished in a single textbook, the
theoretical and modeling topics considered have been organized in two texts; this
text is concerned with rudimentary to advanced theoretical aspects of random ob-
jects based on random variables, including statistical inference, probability theory
and stochastic processes. The modeling of these objects and their applications to
real life data is presented in the text Theory and Modeling of Stochastic Objects:
Point Processes and Random Sets (forthcoming, hereafter referred to as TMSO-
PPRS). The latter stochastic objects are a natural extension of random variables
and vectors and we can think of the TMSO-PPRS text as a natural continuation of
the theory presented herein.