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The present volume contains the transactions of the lOth
Oberwolfach Conference on "Probability Measures on Groups." The
series of these meetings inaugurated in 1970 by L. Schmetterer and
the editor is devoted to an intensive exchange of ideas on a
subject which developed from the relations between various topics
of mathematics: measure theory, probability theory, group theory,
harmonic analysis, special functions, partial differential
operators, quantum stochastics, just to name the most significant
ones. Over the years the fruitful interplay broadened in various
directions: new group-related structures such as convolution
algebras, generalized translation spaces, hypercomplex systems, and
hypergroups arose from generalizations as well as from
applications, and a gradual refinement of the combinatorial,
Banach-algebraic and Fourier analytic methods led to more precise
insights into the theory. In a period of highest specialization in
scientific thought the separated minds should be reunited by
actively emphasizing similarities, analogies and coincidences
between ideas in their fields of research. Although there is no
real separation between one field and another - David Hilbert
denied even the existence of any difference between pure and
applied mathematics - bridges between probability theory on one
side and algebra, topology and geometry on the other side remain
absolutely necessary. They provide a favorable ground for the
communication between apparently disjoint research groups and
motivate the framework of what is nowadays called "Structural
probability theory."
The present volume contains the transactions of the lOth
Oberwolfach Conference on "Probability Measures on Groups." The
series of these meetings inaugurated in 1970 by L. Schmetterer and
the editor is devoted to an intensive exchange of ideas on a
subject which developed from the relations between various topics
of mathematics: measure theory, probability theory, group theory,
harmonic analysis, special functions, partial differential
operators, quantum stochastics, just to name the most significant
ones. Over the years the fruitful interplay broadened in various
directions: new group-related structures such as convolution
algebras, generalized translation spaces, hypercomplex systems, and
hypergroups arose from generalizations as well as from
applications, and a gradual refinement of the combinatorial,
Banach-algebraic and Fourier analytic methods led to more precise
insights into the theory. In a period of highest specialization in
scientific thought the separated minds should be reunited by
actively emphasizing similarities, analogies and coincidences
between ideas in their fields of research. Although there is no
real separation between one field and another - David Hilbert
denied even the existence of any difference between pure and
applied mathematics - bridges between probability theory on one
side and algebra, topology and geometry on the other side remain
absolutely necessary. They provide a favorable ground for the
communication between apparently disjoint research groups and
motivate the framework of what is nowadays called "Structural
probability theory."
Probability measures on algebraic-topological structures such as
topological semi groups, groups, and vector spaces have become of
increasing importance in recent years for probabilists interested
in the structural aspects of the theory as well as for analysts
aiming at applications within the scope of probability theory. In
order to obtain a natural framework for a first systematic
presentation of the most developed part of the work done in the
field we restrict ourselves to prob ability measures on locally
compact groups. At the same time we stress the non Abelian aspect.
Thus the book is concerned with a set of problems which can be
regarded either from the probabilistic or from the
harmonic-analytic point of view. In fact, it seems to be the
synthesis of these two viewpoints, the initial inspiration coming
from probability and the refined techniques from harmonic analysis
which made this newly established subject so fascinating. The goal
of the presentation is to give a fairly complete treatment of the
central limit problem for probability measures on a locally compact
group. In analogy to the classical theory the discussion is
centered around the infinitely divisible probability measures on
the group and their relationship to the convergence of
infinitesimal triangular systems."
By a statistical experiment we mean the procedure of drawing a
sample with the intention of making a decision. The sample values
are to be regarded as the values of a random variable defined on
some meas urable space, and the decisions made are to be functions
of this random variable. Although the roots of this notion of
statistical experiment extend back nearly two hundred years, the
formal treatment, which involves a description of the possible
decision procedures and a conscious attempt to control errors, is
of much more recent origin. Building upon the work of R. A. Fisher,
J. Neyman and E. S. Pearson formalized many deci sion problems
associated with the testing of hypotheses. Later A. Wald gave the
first completely general formulation of the problem of statisti cal
experimentation and the associated decision theory. These achieve
ments rested upon the fortunate fact that the foundations of
probability had by then been laid bare, for it appears to be
necessary that any such quantitative theory of statistics be based
upon probability theory. The present state of this theory has
benefited greatly from contri butions by D. Blackwell and L. LeCam
whose fundamental articles expanded the mathematical theory of
statistical experiments into the field of com parison of
experiments. This will be the main motivation for the ap proach to
the subject taken in this book."
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