Prokhorov and Contemporary Probability Theory: In Honor of Yuri V. Prokhorov

Cover......Page 1
Prokhorov and Contemporary Probability Theory......Page 4
A Conversation with Yuri Vasilyevich Prokhorov......Page 6
Contents......Page 8
Yuri Vasilyevich Prokhorov......Page 12
I Monographs and Textbooks......Page 20
II Main Scientific Papers......Page 21
III Works as Translator and Editor of Translations......Page 28
IV Works as Editor......Page 29
V Prefaces Forewords and Afterwords to the Books......Page 32
VI Book Reviews......Page 33
VII Scientific Life......Page 34
VIII Papers in Encyclopedias......Page 37
1.1 The General Framework: A Rank-Based Reward Game......Page 41
1.2 Methodology......Page 42
1.3.1 The Complete Graph Case......Page 43
1.3.3 Grid with Communication Costs Increasing with Distance......Page 44
1.4 Variant Models and Questions......Page 45
2.1 Finite Number of Rewards......Page 46
2.2 First Passage Percolation : General Setup......Page 47
2.3 First Passage Percolation on the Complete Graph......Page 48
2.4 Analysis of the Rank-Based Reward Game......Page 49
3.1 Nearest-Neighbor First Passage Percolation on the Torus......Page 51
3.2 Analysis of the Rank-Based Reward Game......Page 52
5 The N x N Torus with General Interactions: A Simple Criterion for Efficiency......Page 54
6 The N x N Torus with Short and Long Range Interactions......Page 55
6.1 Order of Magnitude Calculation......Page 56
6.2 First Passage Percolation on the N x N Torus with Short and Long Range Interactions......Page 57
6.3 Exact Equations for the Nash Equilibrium......Page 58
7.1 Transitivity and the Symmetric Variant......Page 61
7.2 Communication at Regular Intervals......Page 62
7.3 Gossip with Reward Based on Audience Size......Page 63
References......Page 64
1 Introduction......Page 65
2 The Basic Model......Page 66
3 Solution to the Investment Timing Problem......Page 68
4 Compensation of Interest Rates by Tax Holidays......Page 72
5 Concluding Remarks......Page 76
References......Page 77
1 Introduction......Page 79
2.1 Definitions and Basics......Page 82
2.2 Meta Distributions......Page 84
3 Light Tails to Light Tails......Page 85
3.1 Sample Clouds......Page 86
3.2 Level Sets and Densities......Page 88
4 Heavy Tails to Heavy Tails......Page 90
4.1 Sample Clouds......Page 92
4.2 Level Sets and Densities......Page 93
5 Heavy Tails to Light Tails......Page 96
6.1 Asymptotic Independence......Page 97
6.2 Asymptotic Dependence and Homothetic Densities......Page 98
7 Conclusion......Page 101
References......Page 102
1 Introduction......Page 104
2 The Setting and the Main Assumptions......Page 105
3 Limit Theorems......Page 109
4.1 The Realised Variation Ratio......Page 116
4.2 The Modified Realised Variation Ratio......Page 118
4.3 Change-of-Frequency Estimator......Page 120
5 Proofs......Page 122
5.1 Proof of Theorem 1......Page 124
5.2 Proof of Theorem 2......Page 127
References......Page 130
1 Introduction and Motivations Stemming from Hidden Processes......Page 131
2 Framework and Main Statements......Page 132
3.1 Proof of Theorem 1......Page 134
3.3 Proof of Proposition 2......Page 136
4 Application to an Identity of Bougerol......Page 139
References......Page 140
1 Introduction......Page 142
2 Appell Polynomials......Page 144
3 Expansion of Convolutions of Measures by Appell Polynomials......Page 145
4 Expansion of a Convolution by Accompanying Probability Measures......Page 149
5 Asymptotic Bergström Expansion......Page 151
6 Expansion of a Convolution by χ2-Distributions......Page 155
References......Page 158
1 Introduction......Page 160
2 The Definitions of Asymptotically Locally Constant Functions. Applications to Limit Theorems on Large Deviations......Page 162
3 The Chracterization of ψ-l.c.f.\'s......Page 165
4 Proofs......Page 168
References......Page 172
1 Introduction......Page 173
2.1 Definitions......Page 174
2.2 Model Description......Page 175
2.2.1 Notation and Preliminary Results......Page 176
2.3.1 Unrestricted Order Sizes......Page 178
2.4 Order Constraints......Page 185
2.5 Sensitivity Analysis......Page 189
3 Conclusion......Page 193
References......Page 194
1 Introduction......Page 195
2 Some Facts About Exponential Levy Models......Page 199
3 Properties of f-Divergence Minimal Martingale Measures......Page 200
4 A Fundamental Equation for f-Divergence Minimal Levy Preserving Martingale Measures......Page 202
4.1 Some Auxiliary Lemmas......Page 203
4.2 A Decomposition for the Density of Levy Preserving Martingale Measures......Page 207
4.3 Proof of Theorem 3 and Proposition 1......Page 212
5.1 First Case: The Interior of supp(ν) Is Not Empty......Page 215
5.2 Second Case: c Is Invertible and ν Is Nowhere Dense......Page 217
5.3 Third Case: c Is Non Invertible and ν Is Nowhere Dense......Page 219
6 Minimal Equivalent Measures When f(x)=axγ......Page 220
6.1 Example......Page 225
References......Page 227
1 Introduction......Page 229
2 A Probabilistic Model for Square-Free Numbers......Page 230
References......Page 244
1 Introduction......Page 246
2 Wiener Maps......Page 248
3 A Refinement......Page 250
4 Concluding Considerations......Page 252
References......Page 254
1 Introduction......Page 255
2 Results......Page 256
3 Auxiliary Results......Page 260
4  A Upper Bound in the Approximation Theorem......Page 264
References......Page 266
1 Introduction......Page 269
2 Sample Correlation Coefficient and Angle Between Vectors......Page 270
3 Some Preliminaries and Remarks......Page 271
4 Main Results......Page 274
5 Bartlett Type Corrections......Page 276
6 Estimates Followed from Approximations for Scale-Mixed Distributions......Page 279
7 Proofs......Page 280
References......Page 289
1 The Stein-Tikhomirov Method and Nonclassical CLT......Page 290
2 Berry-Esseen Inequality for Sampling Sums from Finite Population......Page 297
References......Page 302
References......Page 308
1 Introduction......Page 309
2 Nonlinear Filtering Equations for Superprocesses in Random Environment......Page 311
References......Page 319
1 Markov\'s Bounds for Binomial Coefficients. Preliminaries......Page 321
2 Bounds for b(n,p)......Page 325
References......Page 327
1 Inroduction: Problem and Results......Page 329
2 Proof of Theorem A......Page 332
3 Proof of Theorem B......Page 336
4 Proof of Theorem C......Page 338
5 Proof of Theorem E......Page 339
References......Page 349
1 Introduction......Page 351
3.1 Process Definition and Martingale Representation......Page 352
3.2 Criticality and the Carrying Capacity......Page 354
3.3 Early Extinction......Page 355
3.4 Lingering Around the Carrying Capacity......Page 356
References......Page 357
1 Introduction......Page 359
2 The Statement......Page 360
3 The Proofs......Page 361
4 Previous Work, Remarks, Future Work......Page 370
References......Page 372
1 Introduction......Page 374
2 Exponential Vectors, Weyl Operators, Second Quantization and the Quantum Fourier Transform......Page 375
3 Gaussian States and Their Covariance Matrices......Page 379
4 The Symmetry Group of the Set of Gaussian States......Page 388
References......Page 393
1 Introduction......Page 395
2 Discrete Time......Page 396
3 One-dimensional Diffusion......Page 397
4 Eleven Examples from Dayanik and Karatzas kar......Page 404
5 Other Examples......Page 417
6 Proofs......Page 423
References......Page 425
A Promising Young Man......Page 427
Adventures with Geologists......Page 430
The Academy of Sciences......Page 432
Discrimination......Page 437
How to Predict\rWho Could Travel......Page 439
Meetings Abroad: ICM and Berkeley Symposium......Page 441
The\rWorld Meeting at Tashkent......Page 446
Applied Statistics......Page 448
Kolmogorov......Page 450
The Early Years......Page 453
International Contacts......Page 455
ScientificWork......Page 459
Statistics in the Soviet Union......Page 460
On Discrimination......Page 461