Asymptotic Behaviour Of Linearly Transformed Sums Of Random Variables (mathematics And Its Applications)
by V.V. Buldygin /
1997 / English / PDF
23.2 MB Download
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.
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.
