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1. ARMA Model Identification [1992]
 Choi, ByoungSeon.
 New York, NY : Springer US, 1992.
 Description
 Book — 1 online resource (xii, 200 pages).
 Summary

 1 Introduction. 1.1 ARMA Model. 1.2 History. 1.3 Algorithms. 1.3.1 AR Parameters. 1.3.2 MA Parameters. 1.4 Estimation. 1.4.1 Extended YuleWalker Estimates. 1.4.2 Maximum Likelihood Estimates. 1.5 Nonstationary Processes. 1.5.1 Sample ACRF of a Nonstationary Process. 1.5.2 Iterated Least Squares Estimates. 1.6 Additional References. 2 The Autocorrelation Methods. 2.1 Box and Jenkins' Method. 2.2 The Inverse Autocorrelation Method. 2.2.1 Inverse Autocorrelation Function. 2.2.2 Estimates of the Spectral Density. 2.2.3 Estimates of the IACF. 2.2.4 Identification Using the IACF. 2.3 Additional References. 3 Penalty Function Methods. 3.1 The Final Prediction Error Method. 3.2 Akaike's Information Criterion. 3.2.1 KullbackLeibler Information Number. 3.2.2 Akaike's Information Criterion. 3.3 Generalizations. 3.4 Parzen's Method. 3.5 The Bayesian Information Criterion. 3.5.1 Schwarz' Derivation. 3.5.2 Kashyap's Derivation. 3.5.3 Shortest Data Description. 3.5.4 Some Comments. 3.6 Hannan and Quinn's Criterion. 3.7 Consistency. 3.8 Some Relations. 3.8.1 A Bayesian Interpretation. 3.8.2 The BIC and Prediction Errors. 3.8.3 The AIC and CrossValidations. 3.9 Additional References. 4 Innovation Regression Methods. 4.1 AR and MA Approximations. 4.2 Hannan and Rissanen's Method. 4.2.1 A ThreeStage Procedure. 4.2.2 Block Toeplitz Matrices. 4.2.3 A Modification of the Whittle Algorithm. 4.2.4 Some Modifications. 4.3 Koreisha and Pukkila's Method. 4.4 The KL Spectral Density. 4.5 Additional References. 5 Pattern Identification Methods. 5.1 The 3Pattern Method. 5.1.1 The Three Functions. 5.1.2 Asymptotic Distributions. 5.1.3 Two ChiSquared Statistics Ill. 5.2 The R and S Array Method. 5.2.1 The R and S Patterns. 5.2.2 Asymptotic Distributions. 5.2.3 The RS Array. 5.3 The Corner Method. 5.3.1 Correlation Determinants. 5.3.2 Asymptotic Distribution. 5.4 The GPAC Methods. 5.4.1 Woodward and Gray's GPAC. 5.4.2 Glasbey's GPAC. 5.4.3 Takemura's GPAC. 5.5 The ESACF Method. 5.6 The SCAN Method. 5.6.1 Eigenanalysis. 5.6.2 The SCAN Method. 5.7 Woodside's Method. 5.8 Three Systems of Equations. 5.9 Additional References. 6 Testing Hypothesis Methods. 6.1 Three Asymptotic Test Procedures. 6.2 Some Test Statistics. 6.3 The Portmanteau Statistic. 6.4 Sequential Testing Procedures. 6.5 Additional References.
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2. Aspects of Risk Theory [1991]
 Grandell, Jan.
 New York, NY : Springer New York, 1991.
 Description
 Book — 1 online resource (x, 175 pages 5 illustrations).
 Summary

 1 The classical risk model. 1.1 Ruin probabilities for the classical risk process. 1.2 "Practical" evaluation of ruin probabilities. 1.3 Inference for the risk process. 2 Generalizations of the classical risk model. 2.1 Models allowing for size fluctuation. 2.2 Models allowing for risk fluctuation. 3 Renewal models. 3.1 Ordinary renewal models. 3.2 Stationary renewal models. 3.3 Numerical illustrations. 4 Cox models. 4.1 Markovian intensity: Preliminaries. 4.2 The martingale approach. 4.3 Independent jump intensity. 4.3.1 An inbedded random walk. 4.3.2 Ordinary independent jump intensity. 4.3.3 Stationary independent jump intensity. 4.4 Markov renewal intensity. 4.5 Markovian intensity. 4.5.1 Application of the basic approach. 4.5.2 An alternative approach. 4.6 Numerical illustrations. 5 Stationary models. Appendix. Finite time ruin probabilities. A.1 The classical model. A.2 Renewal models. A.3 Cox models. A.4 Diffusion approximations. References and author index. Inserted surveys. Basic martingale theory. Basic facts about weak convergence. Point processes and martingales. Point processes and random measures. Basic definitions. Superposition of point processes. Thinning of point processes. Basic Markov process theory. Stationary point processes.
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 Anderson, William J.
 New York, NY : Springer New York, 1991.
 Description
 Book — 1 online resource (xii, 355 pages 5 illustrations). Digital: text file; PDF.
 Summary

 1 Transition Functions and Resolvents.
 1. Markov Chains and Transition Functions: Definitions and Basic Properties.
 2. Differentiability Properties of Transition Functions and Significance of the QMatrix.
 3. Resolvent Functions and Their Properties.
 4. The FunctionalAnalytic Setting for Transition Functions and Resolvents.
 5. Feller Transition Functions.
 6. Kendall's Representation of Reversible Transition Functions.
 7. Appendix. 2 Existence and Uniqueness of QFunctions.
 1. QFunctions and the Kolmogorov Backward and Forward Equations.
 2. Existence and Uniqueness of QFunctions. 3 Examples of ContinuousTime Markov Chains.
 1. Finite Markov Chains.
 2. Birth and Death Processes.
 3. Continuous Time Parameter Markov Branching Processes. 4 More on the Uniqueness Problem.
 1. Laplace Transform Tools.
 2. NonuniquenessConstruction of QFunctions Other Than the Minimal One.
 3. UniquenessThe NonConservative Case. 5 Classification of States and Invariant Measures.
 1. Classification of States.
 2. Subinvariant and Invariant Measures.
 3. Classification Based on the QMatrix.
 4. Determination of Invariant Measures from the QMatrix. 6 Strong and Exponential Ergodicity.
 1. The Ergodic Coefficient and Hitting Times.
 2. Ordinary Ergodicity.
 3. Strong Ergodicity.
 4. Geometric Ergodicity for Discrete Time Chains.
 5. The CroftKingman Lemmas.
 6. Exponential Ergodicity for ContinuousTime Chains. 7 Reversibility, Monotonicity, and Other Properties.
 1. Symmetry and Reversibility.
 2. Exponential Families of Transition Functions.
 3. Stochastic Monotonicity and Comparability.
 4. Dual Processes.
 5. Coupling. 8 Birth and Death Processes.
 1. The Potential Coefficients and Feller's Boundary Conditions.
 2. Karlin and McGregor's Representation Theorem and Duality.
 3. The Stieltjes Moment Problem.
 4. The KarlinMcGregor Method of Solution.
 5. Total Positivity of the Birth and Death Transition Function. 9 Population Processes.
 1. Upwardly SkipFree Processes.
 2. Extinction Times and Probability of Extinction for Upwardly SkipFree Processes.
 3. Multidimensional Population Processes.
 4. TwoDimensional Competition Processes.
 5. Birth, Death, and Migration Processes. Symbol Index. Author Index.
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4. ARMA model identification [1992]
 Choi, ByoungSeon.
 New York : SpringerVerlag, c1992.
 Description
 Book — xi, 200 p. ; 24 cm.
 Summary

During the past two decades, considerable progress has been made in statistical time series analysis. The aim of this book is to present a survey of one of the most active areas in this field: the identification of autoregressive movingaverage models, i.e., determining their orders. Readers are assumed to have already taken one course on time series analysis as might be offered in a graduate course, but otherwise this account is selfcontained. The main topics covered include: BoxJenkins' method, inverse autocorrelation functions, penalty function identification such as AIC, BIC techiques and Hannan and Quinn's method, instrumental regression, and a range of pattern identification methods. Rather than cover all the methods in detail, the emphasis is on exploring the fundamental ideas underlying them. References are given to the research literature and as a result, all those engaged, in research in this subject should find this a useful aid to their work.
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QA278.2 .C555 1992  Available 
 Anderson, William J. (William James), 1943
 New York : SpringerVerlag, c1991.
 Description
 Book — xii, 355 p. : ill. ; 25 cm.
 Summary

 Contents: Transition Functions and Resolvents. Existence and Uniqueness of QFunctions. Examples of Continuous Time Markov Chains. More on the Uniqueness Problem. Classification of States and Invariant Measures. Strong and Exponential Ergodicity. Reversibility, Monotonictity, and Other Properties. Birth and Death Processes. Population Processes. Bibliography. Symbol Index. Author Index. Subject Index.
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QA273 .A554 1991  Unknown 
 Durrett, Richard, 1951
 2nd ed.  New York : Springer, ©2008.
 Description
 Book — 1 online resource (xii, 431 pages) : illustrations. Digital: text file; PDF.
 Summary

 Basic models  Estimation and hypothesis testing  Recombination  Population complications  Stepping stone model  Natural selection  Diffusion process  Multidimensional diffusions  Genome rearrangement.
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Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? This book introduces and analyzes a number of probability models: the WrightFisher model, the coalescent, the infinite alleles model, and the infinite sites model.
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 Durrett, Richard, 1951
 2nd ed.  New York : Springer, c2008.
 Description
 Book — xii, 431 p. : ill.
 Summary

 Basic models  Estimation and hypothesis testing  Recombination  Population complications  Stepping stone model  Natural selection  Diffusion process  Multidimensional diffusions  Genome rearrangement.
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Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? This book introduces and analyzes a number of probability models: the WrightFisher model, the coalescent, the infinite alleles model, and the infinite sites model.
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 Del Moral, Pierre.
 New York : SpringerVerlag, c2004.
 Description
 Book — xviii, 555 p. : ill. ; 25 cm.
 Summary

 Introduction. FeynmanKac Formulae. Genealogical and Interacting Particle Models. Stability of FeynmanKac Semigroups. Invariant Measures and Related Topics. Annealing Properties. Asymptotic Behavior. Propagations of Chaos. Central Limit Theorems. Large Deviations Principles. FeynmanKac and Interacting Particle Recipes. Applications.
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QC174.17 .P27 D45 2004  Unknown 
 Silʹvestrov, D. S. (Dmitriĭ Sergeevich)
 London ; New York : Springer, c2004.
 Description
 Book — xiv, 398 p. : ill. ; 24 cm.
 Summary

 Preface. Weak Convergence of Stochastic Processes. Weak Convergence of Randomly Stopped Stochastic Processes. Jconvergence of Compositions of Stochastic Processes. Summary of Applications. Bibliographical Remarks. References. Index.
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QA274 .S548 2004  Available 
10. Foundations of modern probability [2002]
 Kallenberg, Olav.
 2nd ed.  New York : Springer, c2002.
 Description
 Book — xvii, 638 p. ; 24 cm.
 Summary

 Elements of measure theory processes, distributions, and independence random sequences, series, and sums characteristic functions and classical limit theorems conditioning and disintegration Martingales and optional times Markov property and discrete time chains random walk and renewal theory stationarity and ergodic theory Poisson and pure jump type Markov processes Gaussian processes and Brownian motion Skorohod embedding and invariance principles independent increment processes and nullarrays convergence of random processes, measures, and sets stochastic integrals and quadratic variation continuous martingales and Brownian motion Feller processes and semigroups SDEs and martingale problems local time, excursions, and additive functionals onedimensional SDEs and diffusions connections with PDEs and potential theory predictability, compensation, and excessive functions semimartingales and stochastic integration.
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From the reviews of the first edition: "...Kallenberg's present book would have to qualify as the assimilation of probability par excellence. It is a great edifice of material, clearly and ingeniously presented, without any nonmathematical distractions. Readers wishing to venture into it may do so with confidence that they are in very capable hands." F.B. Knight, Mathematical ReviewsThis new edition contains four new chapters as well as numerous improvements throughout the text.
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QA273 .K285 2002  Unknown 
 Durrett, Richard, 1951
 New York : Springer, c2002.
 Description
 Book — viii, 240 p. : ill. ; 25 cm.
 Summary

 Basic Models. Neutral Complications. Natural Selection. Statistical Tests. Genome Rearrangement.
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QH438.4 .S73 D87 2002  Available 
 De la Peña, Víctor.
 New York : Springer, c1999.
 Description
 Book — xv, 392 p. ; 24 cm.
 Summary

 Sums of Independent Random Variables. Randomly Stopped Processes with Independent Increments. Decoupling of UStatistics and UProcesses. Limit Theorems for UStatistics. Limit Theorems for Degenerate UProcesses. General Decoupling Inequalities for Tangent Sequences. Conditionally Independent Sequences. Further Applications of Decoupling.
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QA295 .D355 1999  Unknown 
13. Mass transportation problems [1998]
 Rachev, S. T. (Svetlozar Todorov)
 New York : Springer, c1998.
 Description
 Book — 2 v. : ill. ; 25 cm.
 Summary

 pt. 1. Theory
 pt. 2. Applications.
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QA402.6 .R33 1998 V.1  Unknown 
QA402.6 .R33 1998 V.2  Unknown 
14. Foundations of modern probability [1997]
 Kallenberg, Olav.
 New York : Springer, c1997.
 Description
 Book — xii, 523 p. ; 25 cm.
 Summary

 Elements of measure theory processes, distributions, and independence random sequences, series, and sums characteristic functions and classical limit theorems conditioning and disintegration Martingales and optional times Markov property and discrete time chains random walk and renewal theory stationarity and ergodic theory Poisson and pure jump type Markov processes Gaussian processes and Brownian motion Skorohod embedding and invariance principles independent increment processes and nullarrays convergence of random processes, measures, and sets stochastic integrals and quadratic variation continuous martingales and Brownian motion Feller processes and semigroups SDEs and martingale problems local time, excursions, and additive functionals onedimensional SDEs and diffusions connections with PDEs and potential theory predictability, compensation, and excessive functions semimartingales and stochastic integration.
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QA273 .K285 1997  Unknown 
 Kallenberg, Olav.
 New York : Springer, c1997.
 Description
 Book — xii, 523 p.
 Summary

 Elements of measure theory processes, distributions, and independence random sequences, series, and sums characteristic functions and classical limit theorems conditioning and disintegration Martingales and optional times Markov property and discrete time chains random walk and renewal theory stationarity and ergodic theory Poisson and pure jump type Markov processes Gaussian processes and Brownian motion Skorohod embedding and invariance principles independent increment processes and nullarrays convergence of random processes, measures, and sets stochastic integrals and quadratic variation continuous martingales and Brownian motion Feller processes and semigroups SDEs and martingale problems local time, excursions, and additive functionals onedimensional SDEs and diffusions connections with PDEs and potential theory predictability, compensation, and excessive functions semimartingales and stochastic integration.
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 Galambos, János, 1940
 New York : Springer, c1996.
 Description
 Book — ix, 269 p. ; 24 cm.
 Summary

 Contents: Introduction. The method of polynomials. The Geometric method. The linear programming method. Multivariate Bonferronitype inequalities. Classical problems of probability and applications in combinatorics. Applications in number theory. Statistical applications. Extreme value theory. Miscellaneous topics.
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QA273.6 .G35 1996  Available 
17. The Malliavin calculus and related topics [1995]
 Nualart, David, 1951
 New York : SpringerVerlag, c1995.
 Description
 Book — xi, 266 p. : ill. ; 25 cm.
 Summary

The Malliavin calculus (or stochastic calculus of variations) is an infinitedimensional differential calculus on the Wiener space. Originally, it was developed to prove a probabilistic proof to Hormander's "sum of squares" theorem, but more recently it has found application in a variety of stochastic differential equation problems. This monograph presents the main features of the Malliavin calculus and discusses in detail its connection with the anticipating stochastic calculus. The author begins by developing analysis on the Wiener space, and then uses this to analyze the regularity of probability laws and to prove Hormander's theorem. Subsequent chapters apply the Malliavin calculus to anticipating stochastic differential equations and to studying the Markov property of solutions to stochastic differential equations with boundary conditions.
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QA274.2 .N83 1995  Available 
 Last, Günter.
 New York : SpringerVerlag, c1995.
 Description
 Book — xiv, 490 p. ; 25 cm.
 Summary

This book gives a selfcontained introduction to the dynamic martingale approach to marked point processes (MPP). Based on the notion of a compensator, this approach gives a versatile tool for analyzing and describing the stochastic properties of an MPP. In particular, the authors discuss the relationship of an MPP to its compensator and particular classes of MPP are studied in great detail. The theory is applied to study properties of dependent marking and thinning, to prove results on absolute continuity of point process distributions, to establish sufficient conditions for stochastic ordering between point and jump processes, and to solve the filtering problem for certain classes of MPPs.
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QA274.42 .L37 1995  Unknown 
 Guyon, Xavier.
 New York : SpringerVerlag, 1995.
 Description
 Book — xii, 255 p. : ill. ; 24 cm.
 Summary

The theory of spatial models over lattices, or random fields as they are known, has developed significantly over recent years. This book provides a graduatelevel introduction to the subject which assumes only a basic knowledge of probability and statistics, finite Markov chains, and the spectral theory of secondorder processes. A particular strength of this book is its emphasis on examples  both to motivate the theory which is being developed, and to demonstrate the applications which range from statistical mechanics to image analysis and from statistics to stochastic algorithms.
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QA274.45 .G89 1995  Available 
 Todorovic, P. (Petar)
 New York, NY : Springer New York, 1992.
 Description
 Book — 1 online resource (xiv, 289 pages 15 illustrations).
 Summary

 1 Basic Concepts and Definitions. 1.1. Definition of a Stochastic Process. 1.2. Sample Functions. 1.3. Equivalent Stochastic Processes. 1.4. Kolmogorov Construction. 1.5. Principal Classes of Random Processes. 1.6. Some Applications. 1.7. Separability. 1.8. Some Examples. 1.9. Continuity Concepts. 1.10. More on Separability and Continuity. 1.11. Measurable Random Processes. Problems and Complements. 2 The Poisson Process and Its Ramifications. 2.1. Introduction. 2.2. Simple Point Process on R+. 2.3. Some Auxiliary Results. 2.4. Definition of a Poisson Process. 2.5. Arrival Times ?k. 2.6. Markov Property of N(t) and Its Implications. 2.7. Doubly Stochastic Poisson Process. 2.8. Thinning of a Point Process. 2.9. Marked Point Processes. 2.10. Modeling of Floods. Problems and Complements. 3 Elements of Brownian Motion. 3.1. Definitions and Preliminaries. 3.2. Hitting Times. 3.3. Extremes of ?(t). 3.4. Some Properties of the Brownian Paths. 3.5. Law of the Iterated Logarithm. 3.6. Some Extensions. 3.7. The OrnsteinUhlenbeck Process. 3.8. Stochastic Integration. Problems and Complements. 4 Gaussian Processes. 4.1. Review of Elements of Matrix Analysis. 4.2. Gaussian Systems. 4.3. Some Characterizations of the Normal Distribution. 4.4. The Gaussian Process. 4.5. Markov Gaussian Process. 4.6. Stationary Gaussian Process. Problems and Complements. 5 L2 Space. 5.1. Definitions and Preliminaries. 5.2. Convergence in Quadratic Mean. 5.3. Remarks on the Structure of L2. 5.4. Orthogonal Projection. 5.5. Orthogonal Basis. 5.6. Existence of a Complete Orthonormal Sequence in L2. 5.7. Linear Operators in a Hilbert Space. 5.8. Projection Operators. Problems and Complements. 6 SecondOrder Processes. 6.1. Covariance Function C(s, t). 6.2. Quadratic Mean Continuity and Differentiability. 6.3. Eigenvalues and Eigenfunctions of C(s, t). 6.4. KarhunenLoeve Expansion. 6.5. Stationary Stochastic Processes. 6.6. Remarks on the Ergodicity Property. Problems and Complements. 7 Spectral Analysis of Stationary Processes. 7.1. Preliminaries. 7.2. Proof of the BochnerKhinchin and Herglotz Theorems. 7.3. Random Measures. 7.4. Process with Orthogonal Increments. 7.5. Spectral Representation. 7.6. Ramifications of Spectral Representation. 7.7. Estimation, Prediction, and Filtering. 7.8. An Application. 7.9. Linear Transformations. 7.10. Linear Prediction, General Remarks. 7.11. The Wold Decomposition. 7.12. Discrete Parameter Processes. 7.13. Linear Prediction. 7.14. Evaluation of the Spectral Characteristic ?(?, h). 7.15. General Form of Rational Spectral Density. Problems and Complements. 8 Markov Processes I. 8.1. Introduction. 8.2. Invariant Measures. 8.3. Countable State Space. 8.4. Birth and Death Process. 8.5. Sample Function Properties. 8.6. Strong Markov Processes. 8.7. Structure of a Markov Chain. 8.8. Homogeneous Diffusion. Problems and Complements. 9 Markov Processes II: Application of Semigroup Theory. 9.1. Introduction and Preliminaries. 9.2. Generator of a Semigroup. 9.3. The Resolvent. 9.4. Uniqueness Theorem. 9.5. The HilleYosida Theorem. 9.6. Examples. 9.7. Some Refinements and Extensions. Problems and Complements. 10 Discrete Parameter Martingales. 10.1. Conditional Expectation. 10.2. Discrete Parameter Martingales. 10.3. Examples. 10.4. The Upcrossing Inequality. 10.5. Convergence of Submartingales. 10.6. Uniformly Integrable Martingales. Problems and Complements.
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