Stochastic Processes

25 Aug 2009 22:39

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Things to understand better: Large deviations. Non-asymptotic convergence rates. Convergence properties of non-stationary processes. Coupling methods. Statistical inference on processes. Convergence in distribution of sequences of processes. Empirical process theory (i.e. processes where the index set is a sigma field or a function space), especially when the data are themselves generated by a non-IID process.

A common thing to ask about a stochastic process (assuming it takes values in a vector space) is its time-average. For IID sequences, with finite variance at each time, the ordinary central limit theorem tells us the distribution of the average converges to a Gaussian. With infinite variance, the limiting distribution is one of the Levy distributions. For independent, non-identically distributed sequences, similar but weaker results hold. Quueries: Do we know a necessary and sufficient condition for central limit theorems (Gaussian or Levy) for dependent sequences? (I can find lots of sufficient ones, and trivial necessary ones.) Under some circumstances, one can show that a dependent sequence will converge to an exponential distribution. (The most common example is a random walk with a reflecting barrier.) Do we know necessary and sufficient conditions for convergence to exponentials? (This question is related to the origin of power-law distributions.) Is there a characterization for distributions which can be the limits of averaging dependent random variables? Can we take an IID, finite-variance sequence, and introduce dependence in such a way as to (1) leave the marginal distribution at each time alone but (2) make the limiting distribution Levy? (With thanks to Spyros Skouras for bugging me about these and related matters.)

Markov processes, branching processes and stochastic differential equations are important enough to be spun off into separate notebooks.

See also: Large Deviations; Neural Coding; Nonequilibrium Statistical Mechanics; On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System; Random Time Changes for Stochastic Processes Spatial Statistics and Spatial Stochastic Processes

Recommended, general [including general probability books which I think are especially good on random processes]:
• M. S. Bartlett, An Introduction to Stochastic Processes, with Special Reference to Methods and Applications [Older, far less technical (e.g., no measure theory), but very sound, lots of interesting applications, and a considerable chunk of likelihood-based statistical theory integrated in, which is pleasing.]
• Patrick Billingsley
  • Ergodic Theory and Information
  • Probability and Measure
• Vivek S. Borkar, Probability Theory: An Advanced Course
• J.-R. Chazottes, Books and lecture notes [On probability, stochastic processes, and related subjects]
• Alexandre J. Chorin and Ole Hald, Stochastic Tools in Mathematics and Science
• J. Doob, Stochastic Processes [Since 1953, the Bible.]
• William Feller, An Introduction to the Theory of Probability and Its Applications
• Bert Fristedt and Lawrence Gray, A Modern Approach to Probability Theory [Extremely thorough measure-theoretic text; nice treatment of stochastic processes]
• I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes [Despite the title, this is an advanced book, making extensive use of measure theoretic-probability, which is summarized in one chapter. But it is a very good book of this sort.]
• Robert M. Gray, Probability, Random Processes, and Ergodic Properties [Online]
• Geoffrey Grimmett and David Stirzaker, Probability and Random Processes
• Hoel, Port and Stone, Introduction to Stochastic Processes
• Olav Kallenberg, Foundations of Modern Probability
• R. Lipster, Lecture Notes for Stochastic Processes
• M. Loeve, Probability Theory
• Rinaldo Schinazi, Classical and Spatial Stochastic Processes
• Frank Spitzer, Principles of Random Walk

Recommended, more specialized (very misc.):
• J.-R. Chazottes, P. Collet, C. Kuelske and F. Redig, "Deviation inequalities via coupling for stochastic processes and random fields", math.PR/0503483 [Very cool]
• Harald Cramér, Structural and Statistical Problems for a Class of Stochastic Processes [Very nice results about when stochastic processes can be represented as integrated responses to a driving noise term, with statistical applications in the Gaussian case. Delivered as the first S. S. Wilks lecture at Princeton in 1970, published as a 30 pp. booklet in 1971 by Princeton University Press.]
• Eric Mjolsness, "Stochastic Process Semantics for Dynamical Grammar Syntax: An Overview", cs.AI/0511073
• Robin Pemantle, "A Survey of Random Processes with Reinforcement", math.PR/0610076
• Hernan G. Solari and Mario A. Natiello, "Stochastic population dynamics: The Poisson approximation", Physical Review E 67 (2003): 031918 [PDF]
• Wei Biao Wu, "Nonlinear system theory: Another look at dependence", Proceedings of the National Academy of Sciences 102 (2005): 14150--14154 ["we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms."]

Modesty forbids me to recommend:
• My lecture notes from the 2000 Complex Systems Summer School, for students who had never seen random processes before, or didn't remember them well
• My lecture notes from Statistics 754 at Carnegie Mellon, for students who have had a basic course in measure-theoretic probability and are interested in ergodic theory, large deviations, etc.

To read, teaching:
• Rabi Bhattacharya and Mukul Majumdar, Random Dynamical Systems: Theory and Applications [blurb]
• Adam Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction [Blurb]
• Vincenzo Capasso and David Bakstein, An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine [Blurb]
• Kiyosi Ito, Essentials of Stochastic Processes [blurb. Yes, that Ito.]
• Don S. Lemons, An Introduction to Stochastic Processes in Physics
• Barry L. Nelson, Stochastic Modeling: Analysis and Simulation
• N. G. van Kampen, Stochastic Processes in Physics and Chemistry
• Wax (ed.), Selected Papers on Noise and Stochastic Processes

To read, learning:
• J. M. P. Albin, "On Sampling of stationary increment processes", Annals of Applied Probability 14 (2004): 2016--2037 = math.PR/0403554
• Vitor Araujo, "Random Dynamical Systems", math.DS/0608162 = pp. 330--385 in J.-P. Francoise, G. L. Naber and Tsou S. T. (eds.), Encyclopedia of Mathematical Physics, vol. 3
• V. I. Bakhtin, "Positive Processes", math.DS/0505446 ["we introduce positive flows and processes, which generalize the ordinary dynamical systems and stochastic processes", with promises of laws of large numbers, large deviation properties and action functionals]
• Andrea Baldassarri, Francesca Colaiori and Claudio Castellano, "The average shape of a fluctuation: universality in excursions of stochastic processes," cond-mat/0301068 [A cool result, if true]
• Michele Baldini, "On the Lyapunov Exponent of a Multidimensional Stochastic Flow", math.PR/0610665
• Fulvio Baldovin, Attilio L. Stella, "Central limit theorem for anomalous scaling induced by correlations", cond-mat/0510225
• Nils Berglund and Barbara Gentz, "Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems," cond-mat/0110180
• Alain Boudou, "Approximation of the principal components analysis of a stationary function", Statistics and Probability Letters 76 (2006): 571--578 ["The aim of this article is the approximation of the principal components analysis (PCA) of a stationary function, defined on R, by the PCA of a stationary series. For this, we use a discretization of the real line. We study the behavior of the approximation when the step of discretization decreases."]
• Nicolas Bouleau and Dominique Lépngle, Numerical Methods for Stochastic Process
• Anton Bovier, Statistical Mechanics of Disordered Systems [Blurb]
• Francis Comets, Roberto Fernandez and Pablo A. Ferrari, "Processes with Long Memory: Regenerative Construction and Perfect Simulation," math.PR/0009204
• M. Cranston and Y. LeJean, "Geometric Evolution Under Isotropic Stochastic Flow," Electronic Journal of Probability 3 (1998): 4
• Predrag Cvitanovic, C. P. Dettmann, Ronnie Mainier and Gábor Vattay, "Trace Formulas for Stochastic Evolution Operators: Smooth Conjugation Method," chao-dyn/9811003
• Silvio R. Dahmen, Haye Hinrichsen, Wolfgang Kinzel, "Space Representation of Stochastic Processes with Delay", cond-mat/0703582
• James Davidson, Stochastic Limit Theory: An Introduction for Econometricians
• Victor H. de la Pena, Rustam Ibragimov, and Shaturgun Sharakhmetov, "Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series", math.ST/0611166
• Victor H. De La Pena and Ming Yang, "Bounding the first passage time on an average", Statistics and Probability Letters 67 (2004): 1--7
• J. Dedecker, F. Merlevède, M. Peligrad, "Invariance principles for linear processes. Application to isotonic regression", arxiv:0903.1951 ["maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights these processes can exhibit long range dependence and the limiting distribution is a fractional Brownian motion"]
• Ronald Dickman, "Numerical analysis of the master equation," cond-mat/0110558
• Ronald Dickman and Ronaldo Vidigal
  • "Path Integrals and Perturbation Theory for Stochastic Processes," cond-mat/0205321
  • "Quasi-stationary distributions for stochastic processes with an absorbing state," cond-mat/0110557
• S. H. Djah, H. Gottschalk, and H. Ouerdiane, "Feynman graphs for non-Gaussian measures", math-ph/0501030
• Peter G. Doyle and J. Laurie Snell, "Random Walks and Electric Networks," math.PR/0001057
• Mohamed El Machkouri and Lahcen Ouchti, "Exact convergence rates in the central limit theorem for a class of martingales", math.PR/0403385
• Alison M. Etheridge, An Introduction to Superprocesses [Blurb]
• Sergio Fajardo and H. Jerome Keisler , Model Theory of Stochastic Processes [Review in Bulletin of the London Mathematical Society, reproduced by the publisher]
• David Gamarnik, "Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results," math.PR/0204268
• Tryphon T. Georgiou, "An intrinsic metric for power spectral density functions", math.PR/0608486
• H. Gopalkrishna Gadiyar and R. Padma, "Ramanujan-Fourier series, the Wiener-Khintchine formula and the distribution of prime pairs", Physica A 269 (1999): 503--510 [The autocorrelation function of the primes! (Thanks to Dr. Gadiyar for a reprint)]
• Anne Gegout-Petit and Daniel Commenges, "A general definition of influence between stochastic processes", arxiv:0905.3619
• Beniamin Goldys and Bohdan Maslowski, "Uniform exponential ergodicity of stochastic dissipative systems," math.PR/0111143
• Geoffrey Grimmett and Dominic Welsh, "John Michael Hammersley (1920--2004)", math.PR/0610862
• Vineet Gupta, Radha Jagadeesan and Prakash Panangaden, "Approximate reasoning for real-time probabilistic processes", cs.LO/0505063
• Peter J. Haas, Stochastic Petri Nets: Modelling, Stability, Simulation
• Barry D. Hughes, Random Walks and Random Environments
• Aleksander M. Iksanov, Zbigniew J. Jurek, "Shot noise distributions and selfdecomposability," math.PR/0111069
• L. Ingber, C. Chen, R.P. Mondescu, D. Muzzall and M. Renedo, "Probability tree algorithm for general diffusion processes," physics/0103013
• T. Kuna, J. L. Lebowitz and E. R. Speer, "Realizability of point processes", math-ph/0612075
• Jeffrey C. Lagarias, Eric Rains and Robert J. Vanderbei, "The Kruskal Count," math.PR/0110143
• Liao Ming, Lévy Processes in Lie Groups [blurb]
• Russell Lyons and Jeffrey E. Steif, "Stationary Determinantal Processes: Phase Transitions, Bernoullicity, Entropy, and Domination," math.PR/0204324
• Ashkan Nikeghbali, "A class of remarkable submartingales",
  • "I", math.PR/0505515 ["In this paper, we consider the special class of positive local submartingales $(X_{t})$ of the form: {t}=N_{t}+A_{t}$, where the measure $(dA_{t})$ is carried by the set ${t: X_{t}=0}$. We show that many examples of stochastic processes studied in the literature are in this class..."]
  • "II: Enlargments of filtrations", math.PR/0505623
• Jan Obloj, "The Skorokhod Problem and Its Offspring", math.PR/0401114
• Piero Olla and Luca Pignagnoli, "Local evolution equations for non-Markovian processes", nlin.CD/0502022
• Gilles Pagès, "Quadratic optimal functional quantization of stochastic processes and numerical applications", arxiv:0706.4450 ["Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook."]
• Magda Peligrad and Sergey Utev
  • "A new maximal inequality and invariance principle for stationary sequences", math.PR/0406606 = Annals of Probability 33 (2005): 798--815
  • "Central limit theorem for stationary linear processes", math.PR/0509682 ["We establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in McLeisch (1977) and motivated by Gordin (1969). In doing so we shall preserve the generality of the coefficients, including the long range dependence case, and we shall express the variance of partial sums in a form easy to apply. Ergodicity is not required."]
• Marcus Pivato, "Building a Stationary Stochastic Process From a Finite-dimensional Marginal," math.PR/0108081 [And you thought the Danielli-Kolmogorov Theorem was bad!]
• M. Planat, Noise, Oscillators and Algebraic Randomness: From Noise in Communications Systems to Number Theory
• A. J. Roberts, "Normal form transforms separate slow and fast modes in stochastic dynamical systems", math.DS/0701623
• J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization [Blurb]
• Andrea Rocco and Bruce J. West, "Fractional Calculus and the Evolution of Fractal Phenomena," chao-dyn/9810030
• Ken-Iti Sato, L&eacte;vy Processes and Infinitely Divisible Distributions [blurb]
• Naoki Saito, "The Generalized Spike Process, Sparsity, and Statistical Independence," math.PR/0110103
• S. Satheesh and E. Sandhya, "Semi-Selfdecomposable Laws and Related Processes", math.PR/0412546
• Jacek Serafin, "Finitary Codes, a short survey", math.DS/0608252
• Wojciech Szpankowski, Average Case Analysis of Algorithms on [Preprint version]
• Thorisson, Coupling, Stationarity and Regeneration
• Jiming Yu and Sergio Verdu, "Schemes for Bidirectional Modeling of Discrete Stationary Sources", IEEE Transactions on Information Theory 52 (2006): 4789--4807