Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance

by Samoradnitsky; Gennady

• ISBN13: 9780412051715
• ISBN10: 0412051710
• Edition: 1st
• Format: Hardcover
• Copyright: 1994-06-01
• Publisher: Chapman & Hall/

Summary

Each chapter begins with a brief overview and concludes with a range of exercises at varying levels of difficulty. Proofs are spelled out in detail. DLC: Gaussian processes.

Table of Contents

Preface	xiii
Abbreviations	xvii
Notation	xix
Stable random variables on the real line	1	(54)
Equivalent definitions of a stable distribution	2	(8)
Properties of stable random variables	10	(10)
Symmetric α-stable random variables	20	(1)
Series representation	21	(9)
Series representation of skewed α-stable random variables	30	(5)
Graphs and tables of α-stable densities and c.d.f.'s	35	(6)
Simulation	41	(8)
Exercises	49	(6)
Multivariate stable distributions	55	(56)
Stable random vectors	57	(6)
A counterexample for 0 < α < 1	63	(2)
Characteristic function of an α-stable random vector	65	(7)
Strictly stable and symmetric stable random vectors	72	(5)
Sub-Gaussian random vectors	77	(7)
Complex SαS random variables	84	(3)
Covariation	87	(8)
Covariation norm	95	(2)
James orthogonality	97	(6)
Codifference	103	(4)
Exercises	107	(4)
Stable random processes and stochastic integrals	111	(62)
Stable stochastic processes	112	(1)
Definition of stable integrals as a stochastic process	113	(5)
α-stable random measures	118	(3)
Constructive definition of stable integrals	121	(5)
Properties of stable integrals	126	(9)
Examples	135	(7)
The SαS Levy motion	135	(3)
Moving averages	138	(1)
Ornstein-Uhlenbeck process	138	(1)
Reverse Ornstein-Uhlenbeck process	139	(1)
Well-balanced linear fractional stable motion	140	(1)
Log-fractional stable motion	141	(1)
Real stationary SαS harmonizable process	141	(1)
Sub-Gaussian processes	142	(1)
Sub-stable processes	143	(2)
Series representation for α-stable random measures	145	(4)
A third definition of stable stochastic integrals using the series representation	149	(3)
Condition S	152	(3)
A fourth definition of stable stochastic integrals using a Poisson representation	155	(12)
Exercises	167	(6)
Dependence structures of multivariate stable distributions	173	(50)
Linear regression	174	(7)
Conditional laws that are symmetric around the conditional mean	181	(4)
Linear dependence	185	(2)
Probability tails of order statistics	187	(13)
Joint moments	200	(4)
Association of stable random variables	204	(4)
The codifference for stationary SαS processes	208	(7)
The expected number of level crossings for stationary sub Gaussian processes	215	(2)
Exercises	217	(6)
Non-linear regression	223	(48)
Conditional moments of order greater than or equal to α	224	(12)
Analytic representations of the non-linear regression functions	236	(15)
Examples	251	(4)
Graphical representations	255	(5)
Numerical techniques	260	(10)
Exercises	270	(1)
Complex stable stochastic integrals and harmonizable processes	271	(38)
Complex-valued SαS random measures	272	(3)
Integrals with respect to complex-valued SαS random measures	275	(6)
The complex isotropic SαS case	281	(5)
Series representation of complex-valued SαS random measures and integrals	286	(5)
Harmonizable process	291	(9)
Stationary real harmonizable processes	300	(5)
The codifference for stationary real harmonizable processes	305	(1)
Exercises	306	(3)
Self-similar processes	309	(82)
Self-similarity	311	(7)
Fractional Brownian motion	318	(22)
``Moving average'' representations of fractional Brownian motion	320	(5)
``Harmonizable'' representations of fractional Brownian motion	325	(7)
Fractional Gaussian noise	332	(8)
General characteristics of processes that are α-stable and H-sssi	340	(3)
Linear fractional stable motion	343	(6)
α-stable Levy motion	349	(3)
Log-fractional stable motion	352	(3)
The real harmonizable fractional stable motion	355	(3)
Complex harmonizable fractional stable motion	358	(5)
Subordinated processes	363	(3)
Fractional stable noises	366	(4)
Simulation of fractional noises and motions	370	(6)
ARMA sequences with stable innovations	376	(4)
Fractional ARIMA with stable innovations	380	(7)
Exercises	387	(4)
Chentsov random fields	391	(28)
Self-similar fields with stationary increments in the strong sense	392	(2)
Chentsov random fields	394	(6)
Example: the Levy-Chentsov random field	400	(2)
Example: Takenaka random fields	402	(3)
Properties of Chentsov random fields	405	(2)
Properties of H-sssis Chentsov random fields	407	(3)
Codifference induced by (α, H)-Takenaka fields	410	(4)
Takenaka processes on (0, ∞)	414	(3)
Exercises	417	(2)
Introduction to sample path properties	419	(26)
Versions	420	(1)
Separability	421	(6)
Applications	427	(3)
Measurability	430	(4)
Zero-one laws	434	(5)
Exercises	439	(6)
Boundedness, continuity and oscillations	445	(52)
Introduction	446	(1)
Necessary conditions for sample boundedness	447	(8)
Necessary conditions for sample continuity	455	(5)
Necessary and sufficient conditions for sample boundedness and continuity when 0 < α < 1	460	(10)
Probability tails of suprema of bounded α-stable processes, with index 0 < α < 2	470	(6)
The oscillation process	476	(6)
The case 0 < α < 1	482	(1)
The case 1 ≤ α < 2	483	(1)
The level sets of the oscillation function	484	(2)
A sample path alternative	486	(2)
How strong is the basic assumption?	488	(2)
Exercises	490	(7)
Measurability, integrability and absolute continuity	497	(40)
Existence of a measurable version	498	(4)
Integrability of the sample paths of stable processes	502	(2)
Conditions for integrability	504	(7)
Changing the order of integration	511	(4)
Tail behavior of the LP-norm distribution	515	(4)
Inversion formula for harmonizable SαS processes	519	(5)
Absolute continuity of stable processes	524	(9)
Exercises	533	(4)
Boundedness and continuity via metric entropy	537	(22)
Metric entropy	538	(4)
Sufficient conditions in the case 1 ≤ α 7lt; 2	542	(4)
Necessary conditions in the case 1 ≤ α 7lt; 2	546	(4)
Boundedness and continuity of self-similar α-stable processes	550	(6)
Exercises	556	(3)
Integral representation	559	(12)
Countable parameter space	560	(8)
Arbitrary parameter space	568	(3)
Historical notes and extensions	571	(26)
Notes to Chapter 1	571	(4)
Notes to Chapter 2	575	(2)
Notes to Chapter 3	577	(1)
Notes to Chapter 4	578	(4)
Notes to Chapter 5	582	(3)
Notes to Chapter 6	585	(1)
Notes to Chapter 7	586	(4)
Notes to Chapter 8	590	(2)
Notes to Chapter 9	592	(1)
Notes to Chapter 10	592	(1)
Notes to Chapter 11	593	(1)
Notes to Chapter 12	594	(1)
Notes to Chapter 13	595	(2)
Appendix:	597	(1)
A Table of symmetric α-stable fractiles	597	(6)
Bibliography	603	(18)
Subject index	621	(8)
Author index	629