http://bactra.org/notebooks Cosma's Notebooks en http://bactra.org/notebooks/2012/03/04#stochastic-processes <P>Things to understand better: <a href="large-deviations.html">Large deviations</a>. Non-asymptotic convergence rates. Convergence properties of non-stationary processes. Coupling methods. <a href="time-series.html">Statistical inference</a> on processes. <a href="convergence-of-stochastic-processes.html">Convergence in distribution of sequences of processes</a>. <a href="empirical-process-theory.html">Empirical process theory</a> (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. <P>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 <em>dependent</em> 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 <a href="power-laws.html">power-law distributions</a>.) Is there a characterization for distributions which can be the limits of averaging <em>dependent</em> 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.) <P><a href="markov.html">Markov processes</a>, <a href="branching-processes.html">branching processes</a> and <a href="stoch-diff-eqs.html">stochastic differential equations</a> are important enough to be spun off into separate notebooks. <P>See also: <a href="ergodic-theory.html">Ergodic Theory</a>; <a href="exchangeable.html">Exchangeable Random Sequences; <a href="graph-limits.html">Graph Limits and Infinite Exchangeable Arrays</a>; <a href="large-deviations.html">Large Deviations</a>; <a href="mixing-and-weak-dependence.html">Mixing and Weak Dependence</a>; <a href="neural-coding.html">Neural Coding</a>; <a href="noneq-sm.html">Nonequilibrium Statistical Mechanics</a>; <a href="infinite-stochastic-dyn-sys.html">On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System</a>; <a href="random-time-changes.html">Random Time Changes for Stochastic Processes</a>; <a href="recurrence-times.html">Recurrence Times (also Hitting, Waiting, and First-Passage Times)</a>; <a href="spatial-statistics.html">Spatial Statistics and Spatial Stochastic Processes</a> <ul>Recommended, general [including general probability books which I think are especially good on random processes]: <li>M. S. Bartlett, <cite>An Introduction to Stochastic Processes, with Special Reference to Methods and Applications</cite> [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.] <li>Patrick Billingsley <ul> <li><cite>Ergodic Theory and Information</cite> <li><cite>Probability and Measure</cite> </ul> <li>Vivek S. Borkar, <cite>Probability Theory: An Advanced Course</cite> <li>J.-R. Chazottes, <a href="http://jeanrene.chazottes.googlepages.com/bookslecturenotessurveys">Books and lecture notes</a> [On probability, stochastic processes, and related subjects] <li>Alexandre J. Chorin and Ole Hald, <cite>Stochastic Tools in Mathematics and Science</cite> <li>J. Doob, <cite>Stochastic Processes</cite> [Since 1953, the Bible.] <li>William Feller, <cite>An Introduction to the Theory of Probability and Its Applications</cite> <li>Bert Fristedt and Lawrence Gray, <cite>A Modern Approach to Probability Theory</cite> [Extremely thorough measure-theoretic text; nice treatment of stochastic processes] <li>I. I. Gikhman and A. V. Skorokhod, <cite>Introduction to the Theory of Random Processes</cite> [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 <em>very good</em> book of this sort.] <li>Robert M. Gray, <cite>Probability, Random Processes, and Ergodic Properties</cite> [<a href="http://ee-www.stanford.edu/~gray/arp.html">Online</a>] <li>Geoffrey Grimmett and David Stirzaker, <cite>Probability and Random Processes</cite> <li>Hoel, Port and Stone, <cite>Introduction to Stochastic Processes</cite> <li>Olav Kallenberg, <cite>Foundations of Modern Probability</cite> <li><a href="http://www.eng.tau.ac.il/~liptser/">R. Lipster</a>, <a href="http://www.eng.tau.ac.il/~liptser/list.html">Lecture Notes for Stochastic Processes</a> <li>M. Loeve, <cite>Probability Theory</cite> <li>Rinaldo Schinazi, <cite>Classical and Spatial Stochastic Processes</cite> <li>Frank Spitzer, <cite>Principles of Random Walk</cite> </ul> <ul>Recommended, more specialized (very misc.): <li>Harald Cram&eacute;r, <cite>Structural and Statistical Problems for a Class of Stochastic Processes</cite> [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.] <li>Eric Mjolsness, "Stochastic Process Semantics for Dynamical Grammar Syntax: An Overview", <a href="http://arxiv.org/abs/cs.AI/0511073">cs.AI/0511073</a> <li>Robin Pemantle, "A Survey of Random Processes with Reinforcement", <a href="http://arxiv.org/abs/math.PR/0610076">math.PR/0610076</a> <li>Hernan G. Solari and Mario A. Natiello, "Stochastic population dynamics: The Poisson approximation", <cite>Physical Review E</cite> <strong>67</strong> (2003): 031918 [<a href="http://www.maths.lth.se/matematiklth/personal/mario/talks/stoch.pdf">PDF</a>] </ul> <ul>Modesty forbids me to recommend: <li>My <a href="../prob-notes/">lecture notes from the 2000 Complex Systems Summer School</a>, for students who had never seen random processes before, or didn't remember them well <li>My <a href="http://www.stat.cmu.edu/~cshalizi/754">lecture notes from Statistics 754 at Carnegie Mellon</a>, for students who have had a basic course in measure-theoretic probability and are interested in <a href="ergodic-theory.html">ergodic theory</a>, <a href="large-deviations.html">large deviations</a>, etc. </ul> <ul>To read, teaching: <li>Richard F. Bass, <cite>Stochastic Processes</cite> [<a href="http://cambridge.org/ 9781107008007">Blurb</a>] <li>Rabi Bhattacharya and Mukul Majumdar, <cite>Random Dynamical Systems: Theory and Applications</cite> [<a href="http://cambridge.org/9780521532723">blurb</a>] <li>Adam Bobrowski, <cite>Functional Analysis for Probability and Stochastic Processes: An Introduction</cite> [<a href="http://cambridge.org/0521539374">Blurb</a>] <li>Vincenzo Capasso and David Bakstein, <cite>An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine</cite> [<a href="http://www.springeronline.com/sgw/cda/frontpage/0,11855,5-0-22-34723835-0,00.html">Blurb</a>] <li>Kiyosi Ito, <cite>Essentials of Stochastic Processes</cite> [<a href="http://www.ams.org/bookstore?fn=20&arg1=probability&item=MMONO-231">blurb</a>. Yes, <a href="stoch-diff-eqs.html"><em>that</em> Ito</a>.] <li>Don S. Lemons, <cite>An Introduction to Stochastic Processes in Physics</cite> <li>Barry L. Nelson, <cite>Stochastic Modeling: Analysis and Simulation</cite> <li>N. G. van Kampen, <cite>Stochastic Processes in Physics and Chemistry</cite> <li>Wax (ed.), <cite>Selected Papers on Noise and Stochastic Processes</cite> </ul> <ul>To read, learning: <li>Odd O. Aalen, Per Kragh Andersen, \Ornulf Borgan, Richard D. Gill, Niels Keiding, "History of applications of martingales in survival analysis", <cite>Electronic Journal for History of Probability and Statistics</cite> <strong>5</strong> (2009), <a href="http://arxiv.org/abs/1003.0188">arxiv:1003.0188</a> <li>J. M. P. Albin, "On Sampling of stationary increment processes", <cite>Annals of Applied Probability</cite> <strong>14</strong> (2004): 2016--2037 = <a href="http://arxiv.org/abs/math.PR/0403554">math.PR/0403554</a> <li>Vitor Araujo, "Random Dynamical Systems", <a href="http://arxiv.org/abs/math.DS/0608162">math.DS/0608162</a> = pp. 330--385 in J.-P. Francoise, G. L. Naber and Tsou S. T. (eds.), <cite>Encyclopedia of Mathematical Physics</cite>, vol. 3 <li>V. I. Bakhtin, "Positive Processes", <a href="http://arxiv.org/abs/math.DS/0505446">math.DS/0505446</a> ["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] <li>Andrea Baldassarri, Francesca Colaiori and Claudio Castellano, "The average shape of a fluctuation: universality in excursions of stochastic processes," <a href="http://arxiv.org/abs/cond-mat/0301068">cond-mat/0301068</a> [A cool result, if true] <li>Michele Baldini, "On the Lyapunov Exponent of a Multidimensional Stochastic Flow", <a href="http://arxiv.org/abs/math.PR/0610665">math.PR/0610665</a> <li>Fulvio Baldovin, Attilio L. Stella, "Central limit theorem for anomalous scaling induced by correlations", <a href="http://arxiv.org/abs/cond-mat/0510225">cond-mat/0510225</a> <li>Nicolas Bouleau and Dominique L&eacute;pngle, <cite>Numerical Methods for Stochastic Process</cite> <li>Anton Bovier, <cite>Statistical Mechanics of Disordered Systems</cite> [<a href="http://cambridge.org/0521849918">Blurb</a>] <li>Ga&euml;l Ceillier, "Suficient conditions of standardness for filtrations of stationary processes taking values in a finite space", <a href="http://arxiv.org/abs/1101.1931">arxiv:1101.1931</a> [via a mixing-type condition] <li>Francis Comets, Roberto Fernandez and Pablo A. Ferrari, "Processes with Long Memory: Regenerative Construction and Perfect Simulation," <a href="http://projecteuclid.org/euclid.aoap/1031863175"><cite>Annals of Applied Probability</cite> <strong>12</strong> (2002): 921--943</a>, <a href="http://arxiv.org/abs/math.PR/0009204">math.PR/0009204</a> <li>M. Cranston and Y. LeJean, "Geometric Evolution Under Isotropic Stochastic Flow," <a href="http://www.math.washington.edu/~ejpecp/EjpVol3/paper4.abs.html"><cite>Electronic Journal of Probability</cite> <strong>3</strong> (1998): 4</a> <li>Predrag Cvitanovic, C. P. Dettmann, Ronnie Mainier and G&aacute;bor Vattay, "Trace Formulas for Stochastic Evolution Operators: Smooth Conjugation Method," <a href="http://arxiv.org/abs/chao-dyn/9811003">chao-dyn/9811003</a> <li>Silvio R. Dahmen, Haye Hinrichsen, Wolfgang Kinzel, "Space Representation of Stochastic Processes with Delay", <a href="http://arxiv.org/abs/cond-mat/0703582">cond-mat/0703582</a> <li>James Davidson, <cite>Stochastic Limit Theory: An Introduction for Econometricians</cite> <li>Victor H. de la Pena, Rustam Ibragimov, and Shaturgun Sharakhmetov, "Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series", <a href="http://arxiv.org/abs/math.ST/0611166">math.ST/0611166</a> <li>Moritz Deger, Moritz Helias, Stefano Cardanobile, Fatihcan M. Atay, Stefan Rotter, "Non-equilibrium dynamics of stochastic point processes with refractoriness", <a href="http://arxiv.org/abs/1002.3798">arxiv:1002.3798</a> <li>Ronald Dickman, "Numerical analysis of the master equation," <a href="http://arxiv.org/abs/cond-mat/0110558">cond-mat/0110558</a> <li>Ronald Dickman and Ronaldo Vidigal <ul> <li>"Path Integrals and Perturbation Theory for Stochastic Processes," <a href="http://arxiv.org/abs/cond-mat/0205321">cond-mat/0205321</a> <li>"Quasi-stationary distributions for stochastic processes with an absorbing state," <a href="http://arxiv.org/abs/cond-mat/0110557">cond-mat/0110557</a> </ul> <li>S. H. Djah, H. Gottschalk, and H. Ouerdiane, "Feynman graphs for non-Gaussian measures", <a href="http://arxiv.org/abs/math-ph/0501030">math-ph/0501030</a> <li>Peter G. Doyle and J. Laurie Snell, "Random Walks and Electric Networks," <a href="http://arxiv.org/abs/math.PR/0001057">math.PR/0001057</a> <li>Mohamed El Machkouri and Lahcen Ouchti, "Exact convergence rates in the central limit theorem for a class of martingales", <a href="http://arxiv.org/abs/math.PR/0403385">math.PR/0403385</a> <li>Alison M. Etheridge, <cite>An Introduction to Superprocesses</cite> [<a href="http://www.ams.org/bookstore?fn=50&item=ULECT-20">Blurb</a>] <li>Sergio Fajardo and H. Jerome Keisler , <cite>Model Theory of Stochastic Processes</cite> [Review in <cite>Bulletin of the London Mathematical Society</a>, <a href="http://akpeters.com/pdf/Model_Theory_of_Stochastic_Processes(LNL14)_Bulletin_of_the_LMS.pdf">reproduced</a> by the publisher] <li>Anne Gegout-Petit and Daniel Commenges, "A general definition of influence between stochastic processes", <a href="http://arxiv.org/abs/0905.3619">arxiv:0905.3619</a> <li>Geoffrey Grimmett and Dominic Welsh, "John Michael Hammersley (1920--2004)", <a href="http://arxiv.org/abs/math.PR/0610862">math.PR/0610862</a> <li>Vineet Gupta, Radha Jagadeesan and Prakash Panangaden, "Approximate reasoning for real-time probabilistic processes", <a href="http://arxiv.org/abs/cs.LO/0505063">cs.LO/0505063</a> <li>Peter J. Haas, <cite>Stochastic Petri Nets: Modelling, Stability, Simulation</cite> <li>Michael Hochman, "Upcrossing Inequalities for Stationary Sequences and Applications to Entropy and Complexity", <a href="http://arxiv.org/abs/math.DS/0608311">arxiv:math.DS/0608311</a> [where "complexity" = algorithmic information content] <li>Barry D. Hughes, <cite>Random Walks and Random Environments</cite> <li>Ioannis Kontoyiannis, Sean P. Meyn, "Approximating a Diffusion by a Hidden Markov Model", <a href="http://arxiv.org/abs/0906.0259">arxiv:0906.0259</a> <li>T. Kuna, J. L. Lebowitz and E. R. Speer, "Realizability of point processes", <a href="http://arxiv.org/abs/math-ph/0612075">math-ph/0612075</a> <li>Jeffrey C. Lagarias, Eric Rains and Robert J. Vanderbei, "The Kruskal Count," <a href="http://arxiv.org/abs/math.PR/0110143">math.PR/0110143</a> <li>S. N. Lahiri, "Edgeworth expansions for studentized statistics under weak dependence", <a href="http://projecteuclid.org/euclid.aos/1262271619"><cite>Annals of Statistics</cite> <strong>38</strong> (2010): 388--434</a> <li>Christian Leonard <ul> <li>"Girsanov theory under a finite entropy condition", <a href="http://arxiv.org/abs/110.13958">arxiv:110.13958</a> <li>"Stochastic derivatives and generalized h-transforms of Markov processes", <a href="http://arxiv.org/abs/1102.3172">arxiv:1102.3172</a> </ul> <li>Zenghu Li, <cite>Measure-Valued Branching Markov Processes</cite> [<a href="http://www.springer.com/mathematics/probability/book/978-3-642-15003-6">blurb</a>] <li>Liao Ming, <cite>L&eacute;vy Processes in Lie Groups</cite> [<a href="http://cambridge.org/0521836530">blurb</a>] <li>Russell Lyons and Jeffrey E. Steif, "Stationary Determinantal Processes: Phase Transitions, Bernoullicity, Entropy, and Domination," <a href="http://arxiv.org/abs/math.PR/0204324">math.PR/0204324</a> <li>Ashkan Nikeghbali, "A class of remarkable submartingales", <ul> <li>"I", <a href="http://arxiv.org/abs/math.PR/0505515">math.PR/0505515</a> ["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..."] <li>"II: Enlargments of filtrations", <a href="http://arxiv.org/abs/math.PR/0505623">math.PR/0505623</a> </ul> <li>Jan Obloj, "The Skorokhod Problem and Its Offspring", <a href="http://arxiv.org/abs/math.PR/0401114">math.PR/0401114</a> <li>Piero Olla and Luca Pignagnoli, "Local evolution equations for non-Markovian processes", <a href="http://arxiv.org/abs/nlin.CD/0502022">nlin.CD/0502022</a> <li>Gilles Pag&egrave;s, "Quadratic optimal functional quantization of stochastic processes and numerical applications", <a href="http://arxiv.org/abs/0706.4450">arxiv:0706.4450</a> ["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."] <li>Marc Piegne and Wolfgang Woess, "Stochastic dynamical systems with weak contractivity properties (with a chapter featuring results of Martin Benda)", <a href="http://arxiv.org/abs/1005.2265">arxiv:1005.2265</a> [Results on processes generated by applying an IID sequence of deterministic maps, and so incorporating e.g., state-independent noise perturbing a fixed map.] <li>Magda Peligrad and Sergey Utev <ul> <li>"A new maximal inequality and invariance principle for stationary sequences", <a href="http://arxiv.org/abs/math.PR/0406606">math.PR/0406606</a> = <a href="http://dx.doi.org/10%2E1214/009117904000001035"><cite>Annals of Probability</cite> <strong>33</strong> (2005): 798--815</a> <li>"Central limit theorem for stationary linear processes", <a href="http://arxiv.org/abs/math.PR/0509682">math.PR/0509682</a> ["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."] </ul> <li>Marcus Pivato, "Building a Stationary Stochastic Process From a Finite-dimensional Marginal," <a href="http://arxiv.org/abs/math.PR/0108081">math.PR/0108081</a> [And you thought the Danielli-Kolmogorov Theorem was bad!] <li>M. Planat, <Cite>Noise, Oscillators and Algebraic Randomness: From Noise in Communications Systems to Number Theory</cite> <LI>A. J. Roberts, "Normal form transforms separate slow and fast modes in stochastic dynamical systems", <a href="http://arxiv.org/abs/math.DS/0701623">math.DS/0701623</a> <li>Ken-Iti Sato, <citE>L&eacte;vy Processes and Infinitely Divisible Distributions</cite> [<a href="http://cambridge.org/0521553024">blurb</a>] <li>S. Satheesh and E. Sandhya, "Semi-Selfdecomposable Laws and Related Processes", <a href="http://arxiv.org/abs/math.PR/0412546">math.PR/0412546</a> <li>Jacek Serafin, "Finitary Codes, a short survey", <a href="http://arxiv.org/abs/math.DS/0608252">math.DS/0608252</a> <li>Wojciech Szpankowski, <cite>Average Case Analysis of Algorithms on [<a href="http://www.cs.purdue.edu/homes/spa/book.html">Preprint version</a>] <li>Thorisson, <Cite>Coupling, Stationarity and Regeneration</cite> <li>A. S. Ustunel, <cite>Analysis on Wiener Space and Applications</cite>, <a href="http://arxiv.org/abs/1003.1649">arxiv:1003.1649</a> <li>Ward Whitt, <cite>Stochastic-Process Limits</cite> [<a href="http://www.columbia.edu/~ww2040/book.html">author's website</a>, with selected chapters and a supplement] <li>Jiming Yu and Sergio Verdu, "Schemes for Bidirectional Modeling of Discrete Stationary Sources", <a href="http://dx.doi.org/10.1109/TIT.2006.883626"><cite>IEEE Transactions on Information Theory</cite> <strong>52</strong> (2006): 4789--4807</a> </ul>