Asymptotic Methods in Probability and Statistics With Applications

by Balakrishnan, N.; Ibragimov, I. A.; Nevzorov, Valery B.

• ISBN13: 9780817642143
• ISBN10: 0817642145
• Format: Hardcover
• Copyright: 2001-07-01
• Publisher: Birkhauser

Summary

This book presents thirty-eight extensive and carefully edited chapters written by prominent researchers, providing an up-to-date survey of new asymptotic methods in science, engineering, and technology. The chapters contain broad coverage of recent developments and innovative techniques in a wide range of theoretical and numerical issues in the field of asymptotic methods in probability and mathematical statistics.The book is organized into ten thematic parts: probability distributions; characterizations of distributions; probabilities and measures in high-dimensional structures; weak and strong limit theorems; large deviation probabilities; empirical processes; order statistics and records; estimation of parameters and hypotheses testing; random walks; and applications to finance. Written in an accessible style, this book conveys a clear and practical perspective of asymptotic methods.Features and topics:* Recent developments in asymptotic methods* Parametric and nonparametric inference* Distribution theory* Stochastic processes* Order statistics* Record values and characterizationsAsymptotic Methods in Probability and Statistics with Applications is an essential resource for researchers, practitioners, and professionals involved in theoretical and applied probability and/or in theoretical and applied statistics. Various chapters of the volume will also appeal to industrial statisticians and financial economists.

Table of Contents

Preface	xv
Contributors	xvii
Part I: Probability Distributions
Positive Linnik and Discrete Linnik Distributions	3	(16)
Gerd Christoph
Karina Schreiber
Different Kinds of Linnik's Distributions	3	(3)
Self-deomposability and Discrete Self-decomposability	6	(2)
Scaling of Positive and Discrete Linnik Laws	8	(1)
Strictly Stable and Discrete Stable Distributions as Limit Laws	9	(2)
Asymptotic Expansions	11	(8)
References	15	(4)
On Finite---Dimensional Archimedean Copulas	19	(20)
S. V. Malov
Introduction	19	(3)
Statements of Main Results	22	(3)
Proofs	25	(5)
Some Examples	30	(9)
References	34	(5)
Part II: Characterizations of Distributions
Characterization and Stability Problems for Finite Quadratic Forms	39	(12)
G. Christoph
Yu. Prohorov
V. Ulyanov
Introduction	39	(1)
Notations and Main Results	40	(3)
Auxiliary Results	43	(4)
Proofs of Theorems	47	(4)
References	49	(2)
A Characterization of Gaussian Distributions by Signs of Even Cumulants	51	(4)
L. B. Klebanov
G. J. Szekely
A Conjecture and Main Theorem	51	(2)
An Example	53	(2)
References	53	(2)
On a Class of Pseudo-Isotropic Distributions	55	(10)
A. A. Zinger
Introduction	55	(1)
The Main Results	56	(2)
Proofs	58	(7)
References	61	(4)
Part III: Probabilities and Measures in High-Dimensional Structures
Time Reversal of Diffusion Processes in Hilbert Spaces and Manifolds	65	(16)
Ya. Belopolskaya
Diffusion in Hilbert Space	65	(7)
Duality of time inhomogeneous diffusion processes	69	(3)
Diffusion in Hilbert Manifold	72	(9)
References	79	(2)
Localization of Marjorizing Measures	81	(20)
Bettina Buhler
Wenbo V. Li
Werner Linde
Introduction	81	(2)
Partitions and Weights	83	(1)
Simple Properties of ΘMN (T)	84	(3)
Talagrand's Partitioning Scheme	87	(1)
Majorizing Measures	88	(1)
Approximation Properties	89	(4)
Gaussian Processes	93	(3)
Examples	96	(5)
References	99	(2)
Multidimensional Hungarian Construction for Vectors with Almost Gaussian Smooth Distributions	101	(32)
F. Gotze
A. Yu. Zaitsev
Introduction	101	(5)
The Main Result	106	(6)
Proof of Theorem 8.2.1	112	(11)
Proof of Theorems 8.1.1--8.1.4	123	(10)
References	131	(2)
On the Existence of Weak Solutions for Stochastic Differential Equations With Driving L2-Valued Measures	133	(10)
V. A. Lebedev
Basic Properties of σ-Finite Lp-Valued Random Measures	133	(2)
Formulation and Proof of the Main Result	135	(8)
References	141	(2)
Tightness of Stochastic Families Arising From Randomization Procedures	143	(18)
Mikhail Lifshits
Michel Weber
Introduction	143	(2)
Sufficient Condition of Tightness in C[0, 1]	145	(1)
Continuous Generalization	146	(1)
An Example of Non-Tightness in C[0, 1]	147	(2)
Sufficient Condition for Tightness in Lp[0, 1]	149	(2)
Indicator Functions	151	(4)
An Example of Non-Tightness in Lp, p ∈ [1, 2)	155	(6)
References	158	(3)
Long-Time Behavior of Multi-Particle Markovian Models	161	(16)
A. D. Manita
Introduction	161	(1)
Convergence Time to Equilibrium	162	(1)
Multi-Particle Markov Chains	163	(2)
H and S-Classes of One-Particle Chains	165	(2)
Minimal CTE for Multi-Particle Chains	167	(1)
Proofs	168	(9)
References	176	(1)
Applications of Infinite-Dimensional Gaussian Integrals	177	(12)
A. M. Nikulin
References	187	(2)
On Maximum of Gaussian Non-Centered Fields Indexed on Smooth Manifolds	189	(16)
Vladimir Piterbarg
Sinisha Stamatovich
Introduction	189	(1)
Definitions, Auxiliary Results, Main Results	190	(4)
Proofs	194	(11)
References	203	(2)
Typical Distributions: Infinite-Dimensional Approaches	205	(10)
A. V. Sudakov
V. N. Sudakov
H. v. Weizsacker
Results	205	(10)
References	211	(4)
Part IV: Weak and Strong Limit Theorems
Local Limit Theorem for Stationary Processes in the Domain of Attraction of a Normal Distribution	215	(10)
Jon Aaronson
Manfred Denker
Introduction	215	(1)
Gibbs-Markov Processes and Functionals	216	(2)
Local Limit Theorems	218	(7)
References	223	(2)
On the Maximal Excursion Over Increasing Runs	225	(18)
Andrei Frolov
Alexander Martikainen
Josef Steinebach
Introduction	225	(5)
Results	230	(2)
Proofs	232	(11)
References	240	(3)
Almost Sure Behaviour of Partial Maxima Sequences of Some m-Dependent Stationary Sequences	243	(8)
George Haiman
Lhassan Habach
Introduction	243	(2)
Proof of Theorem 17.1.2	245	(6)
References	249	(2)
On a Strong Limit Theorem for Sums of Independent Random Variables	251	(8)
Valentin V. Petrov
Introduction and Results	251	(2)
Proofs	253	(6)
References	256	(3)
Part V: Large Deviation Probabilities
Development of Linnik's Work in His Investigation of the Probabilities of Large Deviation	259	(18)
A. Aleskeviciene
V. Statulevicius
K. Padvelskis
Reminiscences on Yu. V. Linnik (V. Statulevicius)	259	(1)
Theorems of Large Deviations of Sums of Random Variables Related to a Markov Chain	260	(12)
Non-Gaussian Approximation	272	(5)
References	274	(3)
Lower Bounds on Large Deviation Probabilities for Sums of Independent Random Variables	277	(22)
S. V. Nagaev
Introduction. Statement of Results	277	(6)
Auxiliary Results	283	(3)
Proof of Theorem 20.1.1	286	(5)
Proof of Theorem 20.1.2	291	(8)
References	294	(5)
Part VI: Empirical Processes, Order Statistics, and Records
Characterization of Geometric Distribution Through Weak Records	299	(10)
Fazil A. Aliev
Introduction	299	(1)
Characterization Theorem	300	(9)
References	306	(3)
Asymptotic Distributions of Statistics Based on Order Statistics and Record Values and Invariant Confidence Intervals	309	(12)
Ismihan G. Bairamov
Omer L. Gebizlioglu
Mehmet F. Kaya
Introduction	309	(3)
The Main Results	312	(9)
References	319	(2)
Record Values in Archimedean Copula Processes	321	(12)
N. Balakrishnan
L. N. Nevzorova
V. B. Nevzorov
Introduction	321	(2)
Main Results	323	(4)
Sketch of Proof	327	(6)
References	329	(4)
Functional CLT and LIL for Induced Order Statistics	333	(18)
Yu. Davydov
V. Egorov
Introduction	333	(2)
Notation	335	(1)
Functional Central Limit Theorem	335	(4)
Strassen Balls	339	(4)
Law of the Iterated Logarithm	343	(2)
Applications	345	(6)
References	347	(4)
Notes on the KMT Brownian Bridge Approximation to the Uniform Empirical Process	351	(20)
David M. Mason
Introduction	351	(4)
Proof of the KMT Quantile Inequality	355	(5)
The Diadic Scheme	360	(3)
Some Combinatorics	363	(8)
References	368	(3)
Inter-Record Times in Poisson Paced Fα Models	371	(14)
H. N. Nagaraja
G. Hofmann
Introduction	371	(1)
Exact Distributions	372	(2)
Asymptotic Distributions	374	(11)
References	381	(4)
Part VII: Estimation of Parameters and Hypotheses Testing
Goodness-of-Fit Tests for the Generalized Additive Risk Models	385	(10)
Vilijandas B. Bagdonavicius
Milhail S. Nikulin
Introduction	385	(2)
Test for the First GAR Model Based on the Estimated Score Function	387	(4)
Tests for the Second GAR Model	391	(4)
References	393	(2)
The Combination of the Sign and Wilcoxon Tests for Symmetry and Their Pitman Efficiency	395	(14)
G. Burgio
Ya. Yu. Nikitin
Introduction	395	(2)
Asymptotic Distribution of the Statistic Gn	397	(1)
Pitman Efficiency of the Proposed Statistic	398	(4)
Basic Inequality for the Pitman Power	402	(1)
Pitman Power for Gn	403	(1)
Conditions of Pitman Optimality	404	(5)
References	406	(3)
Exponential Approximation of Statistical Experiments	409	(16)
A. A. Gushchin
E. Valkeila
Introduction	409	(3)
Characterization of Exponential Experiments and Their Convergence	412	(3)
Approximation by Exponential Experiments	415	(10)
References	422	(3)
The Asymptotic Distribution of a Sequential Estimator for the Parameter in an AR(1) Model with Stable Errors	425	(10)
Joop Mijnheer
Introduction	425	(1)
Non-Sequential Estimation	426	(5)
Sequential Estimation	431	(4)
References	433	(2)
Estimation Based on the Empirical Characteristic Function	435	(16)
Bruno Remillard
Radu Theodorescu
Introduction	435	(1)
Tailweight Behavior	436	(2)
Parameter Estimation	438	(5)
An Illustration	443	(3)
Numerical Results and Estimator Efficiency	446	(5)
References	447	(4)
Asymptotic Behavior of Approximate Entropy	451	(14)
Andrew L. Rukhin
Introduction and Summary	451	(2)
Modified Definition of Approximate Entropy and Covariance Matrix for Frequencies	453	(4)
Limiting Distribution of Approximate Entropy	457	(8)
References	460	(5)
Part VIII: Random Walks
Threshold Phenomena in Random Walks	465	(22)
A. V. Nagaev
Introduction	465	(3)
Threshold Phenomena in the Risk Process	468	(1)
Auxiliary Statements	469	(2)
Asymptotic Behavior of the Spitzer Series	471	(7)
The Asymptotic Behavior of M-1	478	(2)
Threshold Properties of the Boundary Functionals	480	(1)
The Limiting Distribution for S	481	(6)
References	484	(3)
Identifying a Finite Graph by Its Random Walk	487	(6)
Heinrich V. Weizsacker
References	490	(3)
Part IX: Miscellanea
The Comparison of the Edgeworth and Bergstrom Expansions	493	(14)
Vladimir I. Chebotarev
Anatolii Ya. Zolotukhin
Introduction and Results	493	(4)
Proof of Lemma 35.1.1	497	(3)
Proof of Lemma 35.1.2	500	(5)
Proof of Theorem 35.1.1	505	(2)
References	505	(2)
Recent Progress in Probabilistic Number Theory	507	(16)
Jonas Kubilius
Results	507	(16)
Part X: Applications to Finance
On Mean Value of Profit for Option Holder: Cases of a Non-Classical and the Classical Market Models	523	(12)
O. V. Rusakov
Notation and Statements	523	(1)
Models	524	(7)
Results	531	(4)
References	533	(2)
On the Probability Models to Control the Investor Portfolio	535	(12)
S. A. Vavilov
Introduction	535	(2)
Portfolio Consisting of Zero Coupon Bonds: The First Scheme	537	(4)
Portfolio Consisting of Arbitrary Securities: The Second Scheme	541	(2)
Continuous Analogue of the Finite-Order Autoregression	543	(2)
Conclusions	545	(2)
References	545	(2)
Index	547