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. Amigo, Matthew B. Kennel and Ljupco Kocarev, "The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems", nlin. Borkar, Mrinal K.
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Ghosh, Ergodic Control of Diffusion Processes Vitor Araujo, "Semicontinuity of entropy, existence of equilibrium states and of physical measures", math. Arnold, Random Dynamical Systems V. Arnol'd and A.
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Thanks to Gustavo Lacerda for the pointer. Chazottes and P. Chazottes, P. Collet and B. Chazottes and G. Gouezel, "On almost-sure versions of classical limit theorems for dynamical systems", math.
Chazottes, G. Chazottes and F. Dedecker, F. Peligrad, "Invariance principles for linear processes. Application to isotonic regression", arxiv Due to the general weights these processes can exhibit long range dependence and the limiting distribution is a fractional Brownian motion"] Thierry De La Rue, "An introduction to joinings in ergodic theory", math. DiMasi and L. We show that the tests of the validity of these assumptions are insufficient by exhibiting observables that are incorrectly sampled with the proposed invariant measures.
Only observables belonging to the 'high temperature phase' of the thermodynamic formalism are insensitive to the sampling methods. We outline methods that are free of these deficiencies.
Eisner, B. Farkas, M.
Haase and R. Constructing orbits with required statistical properties", arxiv Glasner and B. Weiss, "On the interplay between measurable and topological dynamics", math.
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Hairer, "Ergodic properties of a class of non-Markovian processes", arxiv Halmos, Ergodic Theory Bruce E. Hansen ] Nicolai T. Haydn, "The Central Limit Theorem for uniformly strong mixing measures", arxiv Lacroix and Sandro Vaienti, "Hitting and return times in ergodic dynamical systems", math. Horbacz, M. Kregnel, Ergodic Theorems Anna Kuczmaszewska, "The strong law of large numbers for dependent random variables", Statistics and Probability Letters 73 : S.
Shields, "How many future measures can there be? Discrete time systems and flows are covered by this result.
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Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences. Ornstein and B. Petersen, Ergodic Theory Charles Pugh and Michael Shub, with an appendix by Alexander Starkov, "Stable Ergodicity", Bulletin of the American Mathematical Society new series 41 : [ Link ] Maxim Raginsky, "Joint universal lossy coding and identification of stationary mixing sources with general alphabets", arxiv Silva, Invitation to Ergodic Theory Ya. Vilela Mendes, "Beyond Lyapunov", arxiv While it's a relatively young subject, ergodic theory is quite developed.
A crucial feature of ergodic theory is its viewpoint it lends to seemingly unrelated mathematical problems. Ergodic theory has found applications in many parts of mathematics: number theory, harmonic analysis, dynamical systems, probability, mathematical physics. Correspondingly, many recent important developments in mathematics were solved using tools from ergodic theory.
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The importance of ergodic theory in today's mathematics is signified by the fact that some of the greatest prizes in mathematics Fields medal, Wolf prize, Abel prize were given out for work related to ergodic theory. The groups holds weekly research seminars with presentations from faculty and graduate students. The aim is to introduce graduate students to active research areas and guide them to a dissertation topic. For more information on course descriptions, see the graduate catalog. Below are research interests and two representative or recent publications from each member.
Also included for each member is a list of research interests and personal statement. Skip to main content.
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