Asymptotic Methods in Probability and Statistics with Applications

Transcription

1 Asymptotic Methods in Probability and Statistics with Applications N. Balakrishnan LA. Ibragimov V.B. Nevzorov Editors Birkhäuser Boston Basel Berlin

2 Contents Preface Contributors PART I: PROBABILITY DISTRIBUTIONS 1 Positive Linnik and Discrete Linnik Distributions Gerd Christoph and Karina Schreiber 1.1 Different Kinds of Linnik's Distributions Self-deomposability and Discrete Self-decomposability 1.3 Scaling of Positive and Discrete Linnik Laws Strictly Stahle and Discrete Stahle Distributions as Limit Laws Asymptotic Expansions 11 References 15 2 On Finite-Dimensional Archimedean Copulas S. V. Malov 2.1 Introduction Statements of Main Results Proofs Some Examples 30 References 34 PART IL CHARACTERIZATIONS OF DISTRIBUTIONS 3 Characterization and Stability Problems for Finite Quadratic Forms G. Christoph, Yu. Prohorov, and V. Ulyanov 3.1 Introduction Notations and Main Results 40 v

3 vi 3.3 Auxiliary Results Proofs of Theorems 47 References 49 4 A Characterization of Gaussian Distributions by Signs of Even Cumulants L. B. Klebanov and G. J. Szekely 4.1 A Conjecture and Main Theorem An Example 53 References 53 5 On a Class of Pseudo-Isotropic Distributions A. A. Zinger 5.1 Introduction The Main Results Proofs 58 References 61 PART III: PROBABILITIES AND MEASURES IN HlGH-DlMENSIONAL STRUCTURES 6 Time Reversal of Diffusion Processes in Hilbert Spaces and Manifolds Ya. Belopolskaya 6.1 Diffusion in Hilbert Space Duality of time inhomogeneous diffusion processes Diffusion in Hilbert Manifold 72 References 79 7 Localization of Marjorizing Measures Bettina Bühler, Wenbo V. Li, and Werner Linde 7.1 Introduction Partitions and Weights Simple Properties of 6$(T) Talagrand's Partitioning Scheme Majorizing Measures Approximation Properties Gaussian Processes Examples 96 References 99

4 Contents vii 8 Multidimensional Hungarian Construction for Vectors with Almost Gaussian Smooth Distributions 101 F. Götze and A. Yu. Zaitsev 8.1 Introduction The Main Result Proofof Theorem Proof of Theorems References On the Existence of Weak Solutions for Stochastic Differential Equations With Driving L 2 -Valued Measures 133 V. A. Lebedev 9.1 Basic Properties of cr-finite L^-Valued Random Measures Formulation and Proof of the Main Result 135 References Tightness of Stochastic Families Arising From Randomization Procedures 143 Mikhail Lifshits and Michel Weber 10.1 Introduction Sufficient Condition of Tightness in C[0,1] Continuous Generalization An Example of Non-Tightness in C[0,1] Sufficient Condition for Tightness in U>[0,1] Indicator Functions An Example of Non-Tightness in L p, p [1,2) 155 References Long-Time Behavior of Multi-Particle Markovian Models 161 A. D. Manita 11.1 Introduction Convergence Time to Equilibrium Multi-Particle Markov Chains H and S-Classes of One-Particle Chains Minimal CTE for Multi-Particle Chains Proofs 168 References 176

5 viii 12 Applications of Infinite-Dimensional Gaussian Integrals 177 A. M. Nikulin References On Maximum of Gaussian Non-Centered Fields Indexed on Smooth Manifolds 189 Vladimir Piterbarg and Sinisha Stamatovich 13.1 Introduction Definitions, Auxiliary Results, Main Results Proofs 194 References Typical Distributions: Infinite-Dimensional Approaches 205 A. V. Sudakov, V. N. Sudakov, and H. v. Weizsäcker 14.1 Results 205 References 211 PART IV: WEAK AND STRONG LIMIT THEOREMS 15 A Local Limit Theorem for Stationary Processes in the Domain of Attraction of a Normal Distribution 215 Jon Aaronson and Manfred Denker 15.1 Introduction Gibbs-Markov Processes and Functionals Local Limit Theorems 218 References On the Maximal Excursion Over Increasing Runs 225 Andrei Frolov, Alexander Martikainen, and Josef Steinebach 16.1 Introduction Results Proofs 232 References Almost Sure Behaviour of Partial Maxima Sequences of Some m-dependent Stationary Sequences 243 George Haiman and Lhassan Habach 17.1 Introduction Proofof Theorem References 249

6 Contents ix 18 On a Strong Limit Theorem for Sums of Independent Random Variables 251 Valentin V. Petrov 18.1 Introduction and Results Proofs 253 References 256 PART V: LARGE DEVIATION PROBABILITIES 19 Development of Linnik's Work in His Investigation of the Probabilities of Large Deviation 259 A. Aleskeviciene, V. Statulevicius, and K. Padvelskis 19.1 Reminiscences on Yu. V. Linnik (V. Statulevicius) Theorems of Large Deviations of Sums of Random Variables Related to a Markov Chain Non-Gaussian Approximation 272 References Lower Bounds on Large Deviation Probabilities for Sums of Independent Random Variables 277 S. V. Nagaev 20.1 Introduction. Statement of Results Auxiliary Results Proof of Theorem Proofof Theorem References 294 PART VI: EMPIRICAL PROCESSES, ORDER STATISTICS, AND RECORDS 21 Characterization of Geometrie Distribution Through Weak Records 299 Fazil A. Aliev 21.1 Introduction Characterization Theorem 300 References Asymptotic Distributions of Statistics Based on Order Statistics and Record Values and Invariant Confidence Intervals 309 Ismihan G. Bairamov, Omer L. Gebizlioglu, and Mehmet F. Kaya

7 X 22.1 Introduction The Main Results 312 References Record Values in Archimedean Copula Processes 321 N. Balakrishnan, L. N. Nevzorova, and V. B. Nevzorov 23.1 Introduction Main Results Sketch of Proof 327 References Functional CLT and LIL for Induced Order Statistics 333 Yu. Davydov and V. Egorov 24.1 Introduction Notation Functional Central Limit Theorem Strassen Balls Law of the Iterated Logarithm Applications 345 References Notes on the KMT Brownian Bridge Approximation to the Uniform Empirical Process 351 David M. Mason 25.1 Introduction Proof of the KMT Quantile Inequality The Diadic Scheme Some Combinatorics 363 References Inter-Record Times in Poisson Paced F a Models 371 H. N. Nagaraja and G. Hofmann 26.1 Introduction Exact Distributions Asymptotic Distributions 374 References 381 PART VIL ESTIMATION OF PARAMETERS AND HYPOTHESES TESTING 27 Goodness-of-Fit Tests for the Generalized Additive Risk Models 385 Vilijandas B. Bagdonavicius and Milhail S. Nikulin 27.1 Introduction 385

8 Contents xi 27.2 Test for the First GAR Model Based on the Estimated Score Function Tests for the Second GAR Model 391 References The Combination of the Sign and Wilcoxon Tests for Symmetry and Their Pitman Efficiency 395 G. Burgio and Ya. Yu. Nikitin 28.1 Introduction Asymptotic Distribution of the Statistic G n Pitman Efficiency of the Proposed Statistic Basic Inequality for the Pitman Power Pitman Power for G n Conditions of Pitman Optimality 404 References Exponential Approximation of Statistical Experiments 409 A. A. Gushchin and E. Valkeila 29.1 Introduction Characterization of Exponential Experiments and Their Convergence Approximation by Exponential Experiments 415 References The Asymptotic Distribution of a Sequential Estimator for the Parameter in an AR(1) Model with Stable Errors 425 Joop Mijnheer 30.1 Introduction Non-Sequential Estimation Sequential Estimation 431 References Estimation Based on the Empirical Characteristic Function 435 Bruno Remillard and Radu Theodorescu 31.1 Introduction Tailweight Behavior Parameter Estimation An Illustration Numerical Results and Estimator Efficiency 446 References 447

9 xii Contents 32 Asymptotic Behavior of Approximate Entropy 451 Andrew L. Rukhin 32.1 Introduction and Summary Modified Definition of Approximate Entropy and Covariance Matrix for Prequencies Limiting Distribution of Approximate Entropy 457 References 460 PART VIII: RANDOM WALKS 33 Threshold Phenomena in Random Walks 465 A. V. Nagaev 33.1 Introduction Threshold Phenomena in the Risk Process 33.3 Auxiliary Statements Asymptotic Behavior of the Spitzer Series 33.5 The Asymptotic Behavior of M_i Threshold Properties of the Boundary Functionals The Limiting Distribution for S 481 References Identifying a Finite Graph by Its Random Walk 487 Heinrich v. Weizsäcker References 490 PART IX: MISCELLANEA 35 The Comparison of the Edgeworth and Bergström Expansions 493 Vladimir I. Chebotarev and Anatolü Ya. Zolotukhin 35.1 Introduction and Results Proofof Lemma Proofof Lemma Proof of Theorem References Recent Progress in Probabilistic Number Theory 507 Jonas Kubilius 36.1 Results 507

10 Contents xiii PART X: APPLICATIONS TO FINANCE 37 On Mean Value of Profit for Option Holder: Cases of a Non-Classical and the Classical Market Models V. Rusakov 37.1 Notation and Statements Models Results 531 References On the Probability Models to Control the Investor Portfolio 535 S. A. Vavilov 38.1 Introduction Portfolio Consisting of Zero Coupon Bonds: The First Scheine Portfolio Consisting of Arbitrary Securities: The Second Scheme Continuous Analogue of the Finite-Order Autoregression Conclusions 545 References 545 Index 547