Ebook: Asymptotic Behaviour of Linearly Transformed Sums of Random Variables
- Tags: Probability Theory and Stochastic Processes, Statistics general, Sequences Series Summability, Measure and Integration, Systems Theory Control
- Series: Mathematics and Its Applications 416
- Year: 1997
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- 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.
Content:
Front Matter....Pages i-xiii
Front Matter....Pages xv-xv
Random elements and their convergence (preliminary notions)....Pages 1-46
Series of independent random elements....Pages 47-121
Linear transformations of independent random elements and series in sequence spaces....Pages 123-213
Front Matter....Pages 215-215
Operator-normed sums of independent random vectors....Pages 217-268
Operator-normed sums of independent identically distributed random vectors....Pages 269-306
Asymptotic properties of Gaussian Markov sequences....Pages 307-341
Continuity of sample paths of Gaussian Markov processes....Pages 343-362
Asymptotic properties of recurrent random sequences....Pages 363-416
The interplay between strong and weak limit theorems for sums of independent random variables....Pages 417-441
Back Matter....Pages 443-504
Content:
Front Matter....Pages i-xiii
Front Matter....Pages xv-xv
Random elements and their convergence (preliminary notions)....Pages 1-46
Series of independent random elements....Pages 47-121
Linear transformations of independent random elements and series in sequence spaces....Pages 123-213
Front Matter....Pages 215-215
Operator-normed sums of independent random vectors....Pages 217-268
Operator-normed sums of independent identically distributed random vectors....Pages 269-306
Asymptotic properties of Gaussian Markov sequences....Pages 307-341
Continuity of sample paths of Gaussian Markov processes....Pages 343-362
Asymptotic properties of recurrent random sequences....Pages 363-416
The interplay between strong and weak limit theorems for sums of independent random variables....Pages 417-441
Back Matter....Pages 443-504
....