Asymptotic Behaviour of Linearly Transformed Sums of Random Variables

1. Front Matter

   Pages i-xiii

   PDF

2. Random series and linear transformations of sequences of independent random elements

   1. Front Matter

      Pages xv-xv

      PDF

   2. Random elements and their convergence (preliminary notions)

      • Valery Buldygin, Serguei Solntsev

      Pages 1-46

   3. Series of independent random elements

      • Valery Buldygin, Serguei Solntsev

      Pages 47-121

   4. Linear transformations of independent random elements and series in sequence spaces

      • Valery Buldygin, Serguei Solntsev

      Pages 123-213

3. Limit theorems for operator-normed sums of independent random vectors and their applications

   1. Front Matter

      Pages 215-215

      PDF

   2. Operator-normed sums of independent random vectors

      • Valery Buldygin, Serguei Solntsev

      Pages 217-268

   3. Operator-normed sums of independent identically distributed random vectors

      • Valery Buldygin, Serguei Solntsev

      Pages 269-306

   4. Asymptotic properties of Gaussian Markov sequences

      • Valery Buldygin, Serguei Solntsev

      Pages 307-341

   5. Continuity of sample paths of Gaussian Markov processes

      • Valery Buldygin, Serguei Solntsev

      Pages 343-362

   6. Asymptotic properties of recurrent random sequences

      • Valery Buldygin, Serguei Solntsev

      Pages 363-416

   7. The interplay between strong and weak limit theorems for sums of independent random variables

      • Valery Buldygin, Serguei Solntsev

      Pages 417-441

4. Back Matter

   Pages 443-504

   PDF

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.

• Gaussian process
• Markov process
• Probability theory
• Random variable
• mathematical statistics
• statistics

`The book contains a rich amount of new material that can be of interest to many researchers. Numerous sections will be of use to researchers who work with Gaussian processes, stochastic recurrence relations, operator normed sums and weighted sums. Many topics appear in this monographic form for the first time. The excellent bibliographic list contains over 200 titles.! I strongly recommend this book to anyone interested in strong limit theorems.'
kwantitative methoden, 59 (1998)

• Department of Higher Mathematics, Kiev Polytechnic Institute, Kiev, Ukraine

  Valery Buldygin, Serguei Solntsev

Policies and ethics