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Fractional Brownian motion (fBm) is a stochastic process which
deviates significantly from Brownian motion and semimartingales,
and others classically used in probability theory. As a centered
Gaussian process, it is characterized by the stationarity of its
increments and a medium- or long-memory property which is in sharp
contrast with martingales and Markov processes. FBm has become a
popular choice for applications where classical processes cannot
model these non-trivial properties; for instance long memory, which
is also known as persistence, is of fundamental importance for
financial data and in internet traffic. The mathematical theory of
fBm is currently being developed vigorously by a number of
stochastic analysts, in various directions, using complementary and
sometimes competing tools. This book is concerned with several
aspects of fBm, including the stochastic integration with respect
to it, the study of its supremum and its appearance as limit of
partial sums involving stationary sequences, to name but a few. The
book is addressed to researchers and graduate students in
probability and mathematical statistics. With very few exceptions
(where precise references are given), every stated result is
proved.
Fractional Brownian motion (fBm) is a stochastic process which
deviates significantly from Brownian motion and semimartingales,
and others classically used in probability theory. As a centered
Gaussian process, it is characterized by the stationarity of its
increments and a medium- or long-memory property which is in sharp
contrast with martingales and Markov processes. FBm has become a
popular choice for applications where classical processes cannot
model these non-trivial properties; for instance long memory, which
is also known as persistence, is of fundamental importance for
financial data and in internet traffic. The mathematical theory of
fBm is currently being developed vigorously by a number of
stochastic analysts, in various directions, using complementary and
sometimes competing tools. This book is concerned with several
aspects of fBm, including the stochastic integration with respect
to it, the study of its supremum and its appearance as limit of
partial sums involving stationary sequences, to name but a few. The
book is addressed to researchers and graduate students in
probability and mathematical statistics. With very few exceptions
(where precise references are given), every stated result is
proved.
Stein's method is a collection of probabilistic techniques that
allow one to assess the distance between two probability
distributions by means of differential operators. In 2007, the
authors discovered that one can combine Stein's method with the
powerful Malliavin calculus of variations, in order to deduce
quantitative central limit theorems involving functionals of
general Gaussian fields. This book provides an ideal introduction
both to Stein's method and Malliavin calculus, from the standpoint
of normal approximations on a Gaussian space. Many recent
developments and applications are studied in detail, for instance:
fourth moment theorems on the Wiener chaos, density estimates,
Breuer-Major theorems for fractional processes, recursive cumulant
computations, optimal rates and universality results for
homogeneous sums. Largely self-contained, the book is perfect for
self-study. It will appeal to researchers and graduate students in
probability and statistics, especially those who wish to understand
the connections between Stein's method and Malliavin calculus.
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