## Stochastic Processes

*07 Jan 2024 14:15*

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

See also: Cellular Automata; Ergodic Theory; Exchangeable Random Sequences; Graph Limits and Infinite Exchangeable Arrays; Interacting Particle Systems; Large Deviations; Mixing and Weak Dependence; Neural Coding; Nonequilibrium Statistical Mechanics; On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System; Path Integrals and Feynman Diagrams for Classical Stochastic Processes; Prediction Processes; Markovian (and Conceivably Causal) Representations of Stochastic Processes; Projectivity in Statistical Models; Random Fields; Random Time Changes for Stochastic Processes; Recurrence Times (also Hitting, Waiting, and First-Passage Times)

- 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 [Mini-review]
- 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 [Mini-review]
- J. Doob, Stochastic Processes [Mini-review]
- 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 [Mini-review]
- Robert M. Gray, Probability, Random Processes, and Ergodic Properties [Online]
- Geoffrey Grimmett and David Stirzaker, Probability and Random Processes [Mini-review]
- Hoel, Port and Stone, Introduction to Stochastic Processes
- Olav Kallenberg, Foundations of Modern Probability [Mini-review]
- R. Lipster, Lecture Notes for Stochastic Processes
- M. Loeve, Probability Theory [Mini-review]
- Rinaldo Schinazi, Classical and Spatial Stochastic Processes
- Frank Spitzer, Principles of Random Walk

- Recommended, more specialized (very misc.):
- M. S. Bartlett, Stochastic Population Models in Ecology and Epidemiology [Mini-review]
- 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.]
- Vanessa Didelez, "Causal Reasoning for Events in Continuous Time: A Decisionâ€“Theoretic Approach", UAI 2015
- Harry Furstenberg, Stationary Processes and Prediction Theory
- Alexander Gnedin, Ben Hansen, Jim Pitman, "Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws",
Probability Surveys
**4**(2007): 146--171, arxiv:math.PR/0701718 - 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]

- 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
- Almost None of the Theory of Stochastic Process grew from lecture notes for 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:
- Richard F. Bass, Stochastic Processes
- Howard C. Berg, Random Walks in Biology
- Rabi Bhattacharya and Mukul Majumdar, Random Dynamical Systems: Theory and Applications
- Adam Bobrowski, Functional Analysis for Probability and Stochastic Processes: An Introduction
- Denis Bosq and Hung T. Nguyen, A Course in Stochastic Processes: Stochastic Models and Statistical Inference
- Pierre Bremaud, Introduction to Probabilistic Modeling
- Erhan Çinlar, Probability and Stochastics
- Kiyosi Ito, Essentials of Stochastic Processes
- 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

- To read, historical:
- Salomon Bochner, Harmonic Analysis and the Theory of Probability
- Geoffrey Grimmett and Dominic Welsh, "John Michael Hammersley (1920--2004)", math.PR/0610862
- Wax (ed.), Selected Papers on Noise and Stochastic Processes

- To read, learning:
- Odd O. Aalen, Per Kragh Andersen, \Ornulf Borgan, Richard D. Gill, Niels Keiding, "History of applications of martingales in survival analysis",
Electronic Journal for History of Probability and Statistics
**5**(2009), arxiv:1003.0188 - J. M. P. Albin, "On Sampling of stationary increment processes", Annals of Applied Probability
**14**(2004): 2016--2037 = math.PR/0403554 - Giacomo Aletti, Irene Crimaldi, Andrea Ghiglietti, "Networks of reinforced stochastic processes: a complete description of the first-order asymptotics", arxiv:2206.07514
- 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
- Ole E. Barndorff-Nielsen and Albert Shiryaev, Change of Time and Change of Measure
- Patric Bonnier, Harald Oberhauser, "Signature Cumulants, Ordered Partitions, and Independence of Stochastic Processes", arxiv:1908.06496
- Nicolas Bouleau and Dominique Lépngle, Numerical Methods for Stochastic Process
- Gaël Ceillier, "Suficient conditions of standardness for filtrations of stationary processes taking values in a finite space", arxiv:1101.1931 [via a mixing-type condition]
- Francis Comets, Roberto Fernandez and Pablo A. Ferrari, "Processes
with Long Memory: Regenerative Construction and Perfect Simulation," Annals of Applied Probability
**12**(2002): 921--943, 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
- Ronald Dickman, "Numerical analysis of the master equation," cond-mat/0110558
- Ronald Dickman and Ronaldo Vidigal, "Quasi-stationary distributions for stochastic processes with an absorbing state," cond-mat/0110557
- 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
- Sergio Fajardo and H. Jerome Keisler , Model Theory of Stochastic Processes [Review in Bulletin of the London Mathematical Society, reproduced by the publisher]
- Peter Friz and Nicolas B. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications
- Ceillier Gaeël, "Sufficient conditions for the filtration of a stationary processes to be standard", arxiv:1110.5465 [When can you write your stochastic process as a recursive transformation of a sequence of IID noise variables?]
- Anne Gegout-Petit and Daniel Commenges, "A general definition of influence between stochastic processes", arxiv:0905.3619
- 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
- Andreas Hagemann, "Stochastic Equicontinuity in Nonlinear Time Series Models", arxiv:1206.2385
- Martin Hairer, "A theory of regularity structures", arxiv:1303.5113
- Michael Hochman, "Upcrossing Inequalities for Stationary Sequences and Applications to Entropy and Complexity", arxiv:math.DS/0608311 [where "complexity" = algorithmic information content]
- Barry D. Hughes, Random Walks and Random Environments
- Ioannis Kontoyiannis, Sean P. Meyn, "Approximating a Diffusion by a Hidden Markov Model", arxiv:0906.0259
- Dexter Kozen, "Coinductive Proof Principles for Stochastic Processes", arxiv:0711.0194
- Jeffrey C. Lagarias, Eric Rains and Robert J. Vanderbei, "The Kruskal Count," math.PR/0110143
- Henry Lam, "Robust Sensitivity Analysis for Stochastic Systems", arxiv:1303/0326
- Christian Leonard
- "Girsanov theory under a finite entropy condition", arxiv:110.13958
- "Stochastic derivatives and generalized h-transforms of Markov processes", arxiv:1102.3172

- Zenghu Li, Measure-Valued Branching Markov Processes
- Liao Ming, Lévy Processes in Lie Groups
- Russell Lyons and Jeffrey E. Steif, "Stationary Determinantal Processes: Phase Transitions, Bernoullicity, Entropy, and Domination," math.PR/0204324
- Athanasios Christou Micheas, Theory of Stochastic Objects: Probability, Stochastic Processes and Inference
- Ashkan Nikeghbali, "A class of remarkable submartingales",
- "I", math.PR/0505515
- "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
- Marc Piegne and Wolfgang Woess, "Stochastic dynamical systems with weak contractivity properties (with a chapter featuring results of Martin Benda)", arxiv:1005.2265 [Results on processes generated by applying an IID sequence of deterministic maps, and so incorporating e.g., state-independent noise perturbing a fixed map.]
- Magda Peligrad and Sergey Utev
- "A new maximal inequality and
invariance principle for stationary sequences", Annals of
Probability
**33**(2005): 798--815, math.PR/0406606 - "Central limit theorem for stationary linear processes", math.PR/0509682

- "A new maximal inequality and
invariance principle for stationary sequences", Annals of
Probability
- 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
- Ken-Iti Sato, Lévy Processes and Infinitely Divisible Distributions
- 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
- A. S. Ustunel, Analysis on Wiener Space and Applications, arxiv:1003.1649
- Ward Whitt, Stochastic-Process Limits [author's website, with selected chapters and a supplement]
- Jiming Yu and Sergio Verdu, "Schemes for Bidirectional Modeling of
Discrete Stationary
Sources", IEEE
Transactions on Information Theory
**52**(2006): 4789--4807

- To finish writing:
- CRS with Aryeh (Leonid) Kontorovich, Almost None of the Theory of Stochastic Processes