Convergence of Stochastic Processes

25 Jul 2024 09:15

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By which I mean the convergence of sequences of whole processes, i.e., random functions — not the convergence of averages along a process, which is the subject of ergodic theory, and something I understand better. (Of course these two subjects are connected, the bridge being empirical process theory.) I am especially interested in convergence in distribution, a.k.a. weak convergence, though certainly not averse to stronger modes of convergence.

A particularly important class of results are what are called "functional central limit theorems", or "Donsker theorems", or even just "invariance principles". (I hate the last name, but we seem to be stuck with it.) These are all assertions that the processes, appropriately re-scaled, are converging on a fixed limiting Gaussian process, such as the Wiener process or the Brownian bridge. And just as sometimes the central limit theorem for sample averages gives you a Levy distribution rather than a Gaussian, sometimes you get convergence to a Levy process rather than a Gaussian process...

A second important class of results has to do with the convergence of discrete-time, and often discrete-valued, Markov chains to continuous-time Markov processes, either diffusions (which solve stochastic differential equations) or flows (which solve ordinary different equations, i.e., deterministic dynamical systems).

See also:
• Calculating Macroscopic Consequences of Microscopic Interactions
• Large Deviations
• Mean-Field Games and Mean-Field Control
• Stochastic Processes

Recommended:
• Patrick Billingsley, Weak Convergence of Measures: Applications in Probability
• Stewart N. Ethier and Thomas G. Kurtz, Markov Processes: Characterization and Convergence [comments]
• Thomas G. Kurtz
  • Approximation of Population Processes [comments. Like a baby version of Ethier and Kurtz; much easier to read]
  • "Solutions of Ordinary Differential Equations as Limits of Pure Jump Markov Processes", Journal of Applied Probability 7 (1970): 49--58 [JSTOR]
  • "Limit Theorems for Sequences of Jump Markov Processes Approximating Ordinary Differential Processes", Journal of Applied Probability 8 (1971): 344--356 [JSTOR]
  • "Semigroups of Conditioned Shifts and Approximation of Markov Processes", Annals of Probability 3 (1975): 618--642
• I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes
• Olav Kallenberg, Foundations of Modern Probability
• David Pollard, Convergence of Stochastic Processes [Full text free online]
• George G. Roussas, Contiguity of Probability Measures: Some Applications in Statistics

Modesty forbids me to recommend:
• CRS with A. Kontorovich, Almost None of the Theory of Stochastic Processes

To read:
• Patrick Billingsley, Convergence of Probability Measures
• R. W. R. Darling, J. R. Norris, "Differential equation approximations for Markov chains", Probability Surveys 5 (2008): 37--79, arxiv:0710.3269
• Jérôme Dedecker and Sana Louhichi, "Conditional convergence to infinitely divisible distributions with finite variance", Stochastic Processes and Their Applications 115 (2005): 737--768
• Serguei Foss, Takis Konstantopoulos, "A note on the convergence of renewal and regenerative processes to a Brownian bridge", arxiv:0708.3667
• P. E. Greenwood and A. N. Shiryaev, Contiguity and the Statistical Invariance Principle
• J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes
• Hye-Won Kang, Thomas G. Kurtz, Lea Popovic, "Central limit theorems and diffusion approximations for multiscale Markov chain models", arxiv:1208.3783
• Valentin Konakov, Enno Mammen, Jeannette Woerner, "Statistical convergence of Markov experiments to diffusion limits", arxiv:1201.6307
• Ioannis Kontoyiannis, Sean P. Meyn, "Approximating a Diffusion by a Hidden Markov Model", arxiv:0906.0259
• Peter M. Kotelenez and Thomas G. Kurtz, "Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type", Probability Theory and Related Fields 146 (2010): 189--222
• 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
• Ardjen Pengel, Joris Bierkens, "Strong Invariance Principles for Ergodic Markov Processes", arxiv:2111.12603
• Anatoly V. Swishchuk
  • Random Evolutions and Their Applications: New Trends
  • Evolution of Biological Systems in Random Media: Limit Theorems and Stability
• Marta Tyran-Kaminska, "Convergence to Lévy stable processes under strong mixing conditions", arxiv:0907.1185
• Ward Whitt
  • Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues [Ward's site on the book; includes links to PDFs of selected chapters, plus supplements with proofs and errata]
  • "Proofs of the martingale FCLT", arxiv:0712.1929 = Probability Surveys 4 (2007): 268--302