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Volume I of this two-volume text and reference work begins by
providing a foundation in measure and integration theory. It then
offers a systematic introduction to probability theory, and in
particular, those parts that are used in statistics. This volume
discusses the law of large numbers for independent and
non-independent random variables, transforms, special
distributions, convergence in law, the central limit theorem for
normal and infinitely divisible laws, conditional expectations and
martingales. Unusual topics include the uniqueness and convergence
theorem for general transforms with characteristic functions,
Laplace transforms, moment transforms and generating functions as
special examples. The text contains substantive applications, e.g.,
epidemic models, the ballot problem, stock market models and water
reservoir models, and discussion of the historical background. The
exercise sets contain a variety of problems ranging from simple
exercises to extensions of the theory. Volume II of this two-volume
text and reference work concentrates on the applications of
probability theory to statistics, e.g., the art of calculating
densities of complicated transformations of random vectors,
exponential models, consistency of maximum estimators, and
asymptotic normality of maximum estimators. It also discusses
topics of a pure probabilistic nature, such as stochastic
processes, regular conditional probabilities, strong Markov chains,
random walks, and optimal stopping strategies in random games.
Unusual topics include the transformation theory of densities using
Hausdorff measures, the consistency theory using the upper
definition function, and the asymptotic normality of maximum
estimators using twice stochastic differentiability. With an
emphasis on applications to statistics, this is a continuation of
the first volume, though it may be used independently of that book.
Assuming a knowledge of linear algebra and analysis, as well as a
course in modern probability, Volume II looks at statistics from a
probabilistic point of view, touching only slightly on the
practical computation aspects.
What is high dimensional probability? Under this broad name we
collect topics with a common philosophy, where the idea of high
dimension plays a key role, either in the problem or in the methods
by which it is approached. Let us give a specific example that can
be immediately understood, that of Gaussian processes. Roughly
speaking, before 1970, the Gaussian processes that were studied
were indexed by a subset of Euclidean space, mostly with dimension
at most three. Assuming some regularity on the covariance, one
tried to take advantage of the structure of the index set. Around
1970 it was understood, in particular by Dudley, Feldman, Gross,
and Segal that a more abstract and intrinsic point of view was much
more fruitful. The index set was no longer considered as a subset
of Euclidean space, but simply as a metric space with the metric
canonically induced by the process. This shift in perspective
subsequently lead to a considerable clarification of many aspects
of Gaussian process theory, and also to its applications in other
settings.
What is high dimensional probability? Under this broad name we
collect topics with a common philosophy, where the idea of high
dimension plays a key role, either in the problem or in the methods
by which it is approached. Let us give a specific example that can
be immediately understood, that of Gaussian processes. Roughly
speaking, before 1970, the Gaussian processes that were studied
were indexed by a subset of Euclidean space, mostly with dimension
at most three. Assuming some regularity on the covariance, one
tried to take advantage of the structure of the index set. Around
1970 it was understood, in particular by Dudley, Feldman, Gross,
and Segal that a more abstract and intrinsic point of view was much
more fruitful. The index set was no longer considered as a subset
of Euclidean space, but simply as a metric space with the metric
canonically induced by the process. This shift in perspective
subsequently lead to a considerable clarification of many aspects
of Gaussian process theory, and also to its applications in other
settings.
The book is devoted to the fundamental relationship between three
objects: a stochastic process, stochastic differential equations
driven by that process and their associated
Fokker-Planck-Kolmogorov equations. This book discusses wide
fractional generalizations of this fundamental triple relationship,
where the driving process represents a time-changed stochastic
process; the Fokker-Planck-Kolmogorov equation involves
time-fractional order derivatives and spatial pseudo-differential
operators; and the associated stochastic differential equation
describes the stochastic behavior of the solution process. It
contains recent results obtained in this direction.This book is
important since the latest developments in the field, including the
role of driving processes and their scaling limits, the forms of
corresponding stochastic differential equations, and associated FPK
equations, are systematically presented. Examples and important
applications to various scientific, engineering, and economics
problems make the book attractive for all interested researchers,
educators, and graduate students.
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