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Limit theorems for random sequences may conventionally be divided
into two large parts, one of them dealing with convergence of
distributions (weak limit theorems) and the other, with almost sure
convergence, that is to say, with asymptotic prop erties of almost
all sample paths of the sequences involved (strong limit theorems).
Although either of these directions is closely related to another
one, each of them has its own range of specific problems, as well
as the own methodology for solving the underlying problems. This
book is devoted to the second of the above mentioned lines, which
means that we study asymptotic behaviour of almost all sample paths
of linearly transformed sums of independent random variables,
vectors, and elements taking values in topological vector spaces.
In the classical works of P.Levy, A.Ya.Khintchine, A.N.Kolmogorov,
P.Hartman, A.Wintner, W.Feller, Yu.V.Prokhorov, and M.Loeve, the
theory of almost sure asymptotic behaviour of increasing
scalar-normed sums of independent random vari ables was
constructed. This theory not only provides conditions of the almost
sure convergence of series of independent random variables, but
also studies different ver sions of the strong law of large numbers
and the law of the iterated logarithm. One should point out that,
even in this traditional framework, there are still problems which
remain open, while many definitive results have been obtained quite
recently."
It is well known that contemporary mathematics includes many disci
plines. Among them the most important are: set theory, algebra,
topology, geometry, functional analysis, probability theory, the
theory of differential equations and some others. Furthermore,
every mathematical discipline consists of several large sections in
which specific problems are investigated and the corresponding
technique is developed. For example, in general topology we have
the following extensive chap ters: the theory of compact extensions
of topological spaces, the theory of continuous mappings,
cardinal-valued characteristics of topological spaces, the theory
of set-valued (multi-valued) mappings, etc. Modern algebra is
featured by the following domains: linear algebra, group theory,
the theory of rings, universal algebras, lattice theory, category
theory, and so on. Concerning modern probability theory, we can
easily see that the clas sification of its domains is much more
extensive: measure theory on ab stract spaces, Borel and
cylindrical measures in infinite-dimensional vector spaces,
classical limit theorems, ergodic theory, general stochastic
processes, Markov processes, stochastical equations, mathematical
statistics, informa tion theory and many others."
It is well known that contemporary mathematics includes many disci
plines. Among them the most important are: set theory, algebra,
topology, geometry, functional analysis, probability theory, the
theory of differential equations and some others. Furthermore,
every mathematical discipline consists of several large sections in
which specific problems are investigated and the corresponding
technique is developed. For example, in general topology we have
the following extensive chap ters: the theory of compact extensions
of topological spaces, the theory of continuous mappings,
cardinal-valued characteristics of topological spaces, the theory
of set-valued (multi-valued) mappings, etc. Modern algebra is
featured by the following domains: linear algebra, group theory,
the theory of rings, universal algebras, lattice theory, category
theory, and so on. Concerning modern probability theory, we can
easily see that the clas sification of its domains is much more
extensive: measure theory on ab stract spaces, Borel and
cylindrical measures in infinite-dimensional vector spaces,
classical limit theorems, ergodic theory, general stochastic
processes, Markov processes, stochastical equations, mathematical
statistics, informa tion theory and many others."
This book deals with the almost sure asymptotic behaviour of
linearly transformed sequences of independent random variables,
vectors and elements of topological vector spaces. The main
subjects dealing with series of independent random elements on
topological vector spaces, and in particular, in sequence spaces,
as well as with generalized summability methods which are treated
here are strong limit theorems for operator-normed (matrix normed)
sums of independent finite-dimensional random vectors and their
applications; almost sure asymptotic behaviour of realizations of
one-dimensional and multi-dimensional Gaussian Markov sequences;
various conditions providing almost sure continuity of sample paths
of Gaussian Markov processes; and almost sure asymptotic behaviour
of solutions of one-dimensional and multi-dimensional stochastic
recurrence equations of special interest. Many topics, especially
those related to strong limit theorems for operator-normed sums of
independent random vectors, appear in monographic literature for
the first time. Audience: The book is aimed at experts in
probability theory, theory of random processes and mathematical
statistics who are interested in the almost sure asymptotic
behaviour in summability schemes, like operator normed sums and
weighted sums, etc. Numerous sections will be of use to those who
work in Gaussian processes, stochastic recurrence equations, and
probability theory in topological vector spaces. As the exposition
of the material is consistent and self-contained it can also be
recommended as a textbook for university courses.
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