1,732,462 research outputs found
Self-affine sets with positive Lebesgue measure
No full textUsing techniques introduced by C. Gunturk, we prove that the attractors of a family of overlapping self-affine iterated function systems contain a neighbourhood of zero for all parameters in a certain range. This corresponds to giving conditions under which a single sequence may serve as a ‘simultaneous β-expansion’ of different numbers in different bases.Peer reviewe
Ambit stochastics
No full textDrawing on advanced probability theory, Ambit Stochastics is used to model stochastic processes which depend on both time and space. This monograph, the first on the subject, provides a reference for this burgeoning field, complete with the applications that have driven its development. Unique to Ambit Stochastics are ambit sets, which allow the delimitation of space-time to a zone of interest, and ambit fields, which are particularly well-adapted to modelling stochastic volatility or intermittency. These attributes lend themselves notably to applications in the statistical theory of turbulence and financial econometrics. In addition to the theory and applications of Ambit Stochastics, the book also contains new theory on the simulation of ambit fields and a comprehensive stochastic integration theory for Volterra processes in a non-semimartingale context. Written by pioneers in the subject, this book will appeal to researchers and graduate students interested in empirical stochastic modelling
Thermodynamic formalism for suspension flows over countable Markov shifts
No full textWe introduce a new notion of topological pressure for suspension flows over countable Markov shifts. We show that our definition, which is a natural analogue of Gurevich pressure for a Markov shift, is equivalent to previous notions of topological pressure which were defined on a restricted class of flows. We also widen the domain of definition of these previous notions, making them suitable for application to some important examples
Regeneration in random combinatorial structures
Get PDFKingman’s theory of partition structures relates, via a natural sampling procedure, finite partitions to hypothetical infinite populations. Explicit formulas for distributions of such partitions are rare, the most no- table exception being the Ewens sampling formula, and its two-parameter extension by Pitman. When one adds an extra structure to the partitions like a linear order on the set of blocks and regenerative properties, some representation theorems allow to get more precise information on the dis- tribution. In these notes we survey recent developments of the theory of regenerative partitions and compositions. In particular, we discuss connec- tion between ordered and unordered structures, regenerative properties of the Ewens-Pitman partitions, and asymptotics of the number of compo- nents
Counting β-expansions and the absolute continuity of Bernoulli convolutions
No full textWe study the typical growth rate of the number of words of length n which can be extended to β-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter β is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions
A species sampling model with finitely many types
No full textA two-parameter family of exchangeable partitions with a simple updating rule is introduced. The partition is identified with a randomized version of a standard symmetric Dirichlet speciessampling model with finitely many types. A power-like distribution for the number of types is derived
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