Asymptotic behaviour of linearly transformed sums of random variables:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Dordrecht u.a.
Kluwer
1997
|
| Series: | Mathematics and its applications
416 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 500 S. |
| ISBN: | 0792346327 |
Staff View
MARC
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| 245 | 1 | 0 | |a Asymptotic behaviour of linearly transformed sums of random variables |c by Valery Buldygin and Serguei Solntsev |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer |c 1997 | |
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Record in the Search Index
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| adam_text | Contents
Preface ix
Part I Random series and linear transformations of
sequences of independent random elements 1
Chapter 0 Random elements and their convergence
(preliminary notions) 1
0.1 F spaces and separating sets of functional 1
0.1.1 Topological vector spaces 1
0.1.2 Separating sets of functionals and weak topologies 3
0.1.3 F spaces , 5
0.1.4 Classes of topologies on F spaces 7
0.1.5 The space R 7
0.1.6 Linear operators and matrices 8
0.2 cr algebras and measures. Convergence of measures 11
0.2.1 (T algebras 11
0.2.2 Pre measures, measures and characteristic functionals 13
0.2.3 Weak convergence of measures 16
0.2.4 T weak and essentially weak convergence 17
0.3 Random elements and their characteristics 19
0.3.1 Random elements 19
0.3.2 Distributions of random elements 20
0.3.3 Mean values and characteristic functionals 20
0.3.4 Covariance characteristics 21
0.3.5 Independent random elements 22
0.4 Convergence of random elements 24
0.4.1 Almost sure convergence 24
0.4.2 Convergence in probability 25
0.4.3 Convergence in distribution 25
0.4.4 T weak and essentially weak almost sure convergence 27
0.5 Sums of independent random elements 31
0.5.1 Inequalities for sums 31
0.5.2 The weak law of large numbers for sums of independent
random variables , 35
v
vi
0.6 Gaussian random elements 36
0.6.1 Gaussian random variables 36
0.6.2 Gaussian random vectors 39
0.6.3 Gaussian random elements 41
Chapter 1 Series of independent random elements 47
1.1 The symmetrization principle. Four remarkable theorems on series of
independent random variables 47
1.2 The Levy theorem in F spaces 53
1.3 Equivalence of the strong and weak almost sure convergence of series
of independent symmetric summands 55
1.4 Weak topologies and convergence of series of independent symmetric
summands 60
1.5 Fourier analysis and convergence of series of independent terms in
Hilbert spaces 65
1.6 Series with stable terms in Hilbert spaces 77
1.7 Integrability of sums of independent random elements 83
1.8 The Abel transformation and the contraction principle for random
series 92
1.9 The majorization principle for random series 96
1.10 Sub Gaussian random variables. Gaussian majorization of
sub Gaussian series 99
1.11 Random series in the space of continuous functions 110
Chapter 2 Linear transformations of independent random
elements and series in sequence spaces 123
2.1 Random elements in sequence spaces 124
2.2 Linear summability schemes and series in sequence spaces 137
2.3 Stochastic arrays and linear sequences. Oscillation properties of linear
sequences 145
2.4 Oscillation properties of Gaussian sequences 161
2.5 Multiplicative transformations of stochastic arrays. Examples .... 175
2.6 The contraction principle for stochastic arrays 180
2.7 Strong laws of large numbers for weighted sums of independent sum¬
mands 194
2.8 Generalized summability methods 197
2.9 Stability in probability of linear summability schemes 201
2.10 Gaussian majorization for linear transformations of independent sub
Gaussian random variables and vectors 206
Part II Limit theorems for operator normed sums
of independent random vectors and their applications 215
Chapter 3 Operator normed sums of independent random vectors 217
3.1 The Prokhorov Loeve type strong laws of large numbers 218
vii
! 3.2 Strong laws of large numbers for operator normed sums of
independent random vectors 225
3.3 Strong laws of large numbers for spherically symmetric random
vectors 244
3.4 Almost sure boundedness and the iterated logarithm type laws .... 248
3.5 Almost sure convergence of operator normed sums of independent
random vectors 254
3.6 Operator normed sums of independent Gaussian and sub Gaussian
vectors 261
Chapter 4 Operator normed sums of independent identically
distributed random vectors 269
4.1 Integral type criteria 270
4.2 Some properties of sums of independent identically distributed
random vectors with finite second moments 281
4.3 The equivalence of operator and scalar normalizations for sums of
independent identically distributed random vectors with finite second
moments. Integral criteria 286
4.4 Strong relative stability of linearly transformed sums of independent
identically distributed symmetric random vectors 298
Chapter 5 Asymptotic properties of Gaussian Markov sequences 307
5.1 Gaussian Markov sequences and stochastic recurrence equations . . . 307
5.2 Entropy conditions of boundedness and convergence of Gaussian
Markov sequences 325
5.3 One dimensional Gaussian Markov sequences 332
Chapter 6 Continuity of sample paths of Gaussian Markov
processes 343
6.1 Oscillations of Gaussian processes 343
6.2 The equivalence of sample and sequential continuity of Gaussian
processes 349
6.3 A rank criterion of continuity of Gaussian Markov processes 354
6.4 An entropy criterion of continuity of Markov processes 361
Chapter 7 Asymptotic properties of recurrent random sequences 363
7.1 Convergence to zero of Gaussian Markov sequences in Rm 364
7.2 A Gaussian majorization principle for solutions of stochastic
recurrence equations with sub Gaussian perturbations 368
7.3 Almost sure convergence to zero of rra th order random recurrent
: sequences in R 373
7.4 Almost sure boundedness and the iterated logarithm type laws for
I normalized rn th order recurrent sequences in R 388
• 7.5 Asymptotic behaviour of recurrent sequences in Rm 395
7.6 Strong laws of large numbers and the iterated logarithm type laws
for sums of elements of recurrent sequences in Hm (m 1) 397
f
viii
7.7 Appendix. Inequalities for the norms of the matrices AnH 410
Chapter 8 The interplay between strong and weak limit theorems
for sums of independent random variables 417
8.1 A characterization of the law of the iterated logarithm in terms of
asymptotic normality 418
8.2 UDPA and UNA: two special classes of sequences of random
variables 424
8.3 Normalization and strong relative stability of sums of UDPA random
variables 431
8.4 Strong and weak limit theorems for UNA random variables 432
8.5 Normalization and strong relative stability of weighted sums of
independent identically distributed random variables 436
Comments 443
Bibliography 455
Subject index 487
List of Notations 495
|
| any_adam_object | 1 |
| author | Buldygin, V. V. 1946- Solncev, Sergej |
| author_GND | (DE-588)105069600X |
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012179604 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:23:08Z |
| institution | BVB |
| isbn | 0792346327 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008253118 |
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| physical | XIII, 500 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Buldygin, V. V. 1946- Verfasser (DE-588)105069600X aut Asymptotic behaviour of linearly transformed sums of random variables by Valery Buldygin and Serguei Solntsev Dordrecht u.a. Kluwer 1997 XIII, 500 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 416 Limit theorems (Probability theory) Random variables Summability theory Asymptotik (DE-588)4126634-1 gnd rswk-swf Reihe (DE-588)4049197-3 gnd rswk-swf Zufallsvektor (DE-588)4191098-9 gnd rswk-swf Zufallsvektor (DE-588)4191098-9 s Reihe (DE-588)4049197-3 s Asymptotik (DE-588)4126634-1 s DE-604 Solncev, Sergej Verfasser aut Mathematics and its applications 416 (DE-604)BV008163334 416 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008253118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Buldygin, V. V. 1946- Solncev, Sergej Asymptotic behaviour of linearly transformed sums of random variables Mathematics and its applications Limit theorems (Probability theory) Random variables Summability theory Asymptotik (DE-588)4126634-1 gnd Reihe (DE-588)4049197-3 gnd Zufallsvektor (DE-588)4191098-9 gnd |
| subject_GND | (DE-588)4126634-1 (DE-588)4049197-3 (DE-588)4191098-9 |
| title | Asymptotic behaviour of linearly transformed sums of random variables |
| title_auth | Asymptotic behaviour of linearly transformed sums of random variables |
| title_exact_search | Asymptotic behaviour of linearly transformed sums of random variables |
| title_full | Asymptotic behaviour of linearly transformed sums of random variables by Valery Buldygin and Serguei Solntsev |
| title_fullStr | Asymptotic behaviour of linearly transformed sums of random variables by Valery Buldygin and Serguei Solntsev |
| title_full_unstemmed | Asymptotic behaviour of linearly transformed sums of random variables by Valery Buldygin and Serguei Solntsev |
| title_short | Asymptotic behaviour of linearly transformed sums of random variables |
| title_sort | asymptotic behaviour of linearly transformed sums of random variables |
| topic | Limit theorems (Probability theory) Random variables Summability theory Asymptotik (DE-588)4126634-1 gnd Reihe (DE-588)4049197-3 gnd Zufallsvektor (DE-588)4191098-9 gnd |
| topic_facet | Limit theorems (Probability theory) Random variables Summability theory Asymptotik Reihe Zufallsvektor |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008253118&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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