Dynamical Systems and Processes (Irma Lectures in Mathematics and Theoretical Physics)

Preface......Page 5
Contents......Page 9
I Spectral theorems and convergence in mean......Page 13
Bochner–Herglotz lemma......Page 15
The spectral inequality......Page 20
The von Neumann theorem......Page 22
The spectral regularization inequality......Page 38
Moving averages......Page 56
Uniform distribution mod a – the Weyl criterion......Page 63
The van der Corput principle......Page 67
Weakly stationary processes......Page 73
Spectral representation of unitary operators......Page 76
Elements of stochastic integration......Page 88
Spectral representation of weakly stationary processes......Page 90
Weakly stationary sequences and orthogonal series......Page 92
Gaposhkin\'s spectral criterion......Page 97
II Ergodic Theorems......Page 103
Measurable dynamical systems – topological dynamical systems......Page 105
Ergodicity of a dynamical system......Page 113
Weak mixing, strong mixing, continuous spectrum......Page 115
Spectral mixing theorem......Page 121
Other equivalences and other forms of mixing......Page 126
Examples......Page 133
Birkhoff\'s pointwise theorem......Page 141
Dominated ergodic theorems......Page 151
Classes  L logm L......Page 156
A converse......Page 157
Speed of convergence......Page 160
Oscillation functions of ergodic averages......Page 164
Wiener–Wintner theorem......Page 177
Weighted ergodic averages......Page 180
Subsequence averages......Page 205
Banach principle......Page 212
Continuity principle......Page 218
Applications......Page 229
A principle of domination – conjugacy lemma......Page 238
Some liaison theorems......Page 242
Two preliminary lemmas......Page 254
Proof of Theorem 6.1.1......Page 259
Proof of Theorem 6.1.6......Page 261
The case Lp, 1A remarkable GB set property......Page 271
Introduction and preliminaries......Page 279
A theorem of Burton and Denker......Page 281
The central limit theorem for orbits......Page 296
A theorem of Volný......Page 301
CLT for rotations......Page 303
Lacunary series and convergence in variation......Page 327
III Methods arising from the theory of stochastic processes......Page 351
Introduction and general results......Page 353
A theorem of Stechkin......Page 361
An application to the quantitative Borel–Cantelli lemma......Page 365
Application to Gál–Koksma\'s theorems......Page 376
An application to the supremum of random polynomials......Page 381
Application to a.s. convergence of weighted series of contractions......Page 399
An application to random perturbation of intersective sets......Page 415
An application to the discrepancy of some random sequences......Page 421
An application to random Dirichlet polynomials......Page 427
Introduction – the exponential case......Page 445
A general approach......Page 450
A useful criterion......Page 459
Proof of Theorem 9.3.3......Page 469
Proof of Theorems 9.3.10 and 9.3.11......Page 481
Proof of Theorem 9.3.12 and some examples......Page 483
A stronger form of Salem–Zygmund\'s estimate......Page 487
Some examples and discussion......Page 490
Uniform convergence of random Fourier series......Page 500
Gaussian variables and correlation estimates......Page 503
0-1 laws, integrability and comparison lemmas......Page 516
Regularity and irregularity of Gaussian processes......Page 522
Gaussian suprema......Page 529
Oscillations of Gaussian Stein\'s elements......Page 541
Tightness of Gaussian Stein\'s elements......Page 549
IV Three studies......Page 559
Introduction......Page 561
The results of Jessen and Rudin......Page 563
Individual theorems of spectral type......Page 566
Breadth and dimension......Page 569
Bourgain\'s results......Page 574
Connection with number theory......Page 577
Riemann sums and the randomly sampled trigonometric system......Page 585
Almost sure convergence and square functions of Riemann sums......Page 599
Introduction and mean convergence......Page 613
Almost sure convergence – sufficient conditions......Page 623
Almost sure convergence – necessary conditions......Page 646
Random sequences......Page 654
Introduction......Page 671
Value distribution and small divisors of Bernoulli sums......Page 673
An LIL for arithmetic functions......Page 687
On the order of magnitude of the divisor functions......Page 697
Value distribution of the divisors of n2+1......Page 703
Value distribution of the divisors of Rademacher sums......Page 711
The functional equation and the Lindelöf Hypothesis......Page 713
An extremal divisor case......Page 723
Bibliography......Page 741
Index......Page 771