On typical encodings of multivariate ergodic sources
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Anotace
We show that the typical coordinate-wise encoding of multivariate ergodic source into prescribed alphabets has the entropy profile close to the convolution of the entropy profile of the source and the modular polymatroid that is determined by the cardinalities of the output alphabets. We show that the proportion of the exceptional encodings that are not close to the convolution goes to zero doubly exponentially. The result holds for a class of multivariate sources that satisfy asymptotic equipartition property described via the mean fluctuation of the information functions. This class covers asymptotically mean stationary processes with ergodic mean, ergodic processes, irreducible Markov chains with an arbitrary initial distribution. We also proved that typical encodings yield the asymptotic equipartition property for the output variables. These asymptotic results are based on an explicit lower bound of the proportion of encodings that transform a multivariate random variable into a variable with the entropy profile close to the suitable convolution.
Support of solutions of stochastic differential equations in exponential Besov–Orlicz spaces
Autoři
Rok
2018
Publikováno
Stochastic Analysis and Applications. 2018, 2018(6), ISSN 0736-2994.
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The Besov–Orlicz space B1/2Φ,∞(0,T;Rd) with Φ(x)= exp(x2)−1 is currently the smallest known classical function space to which paths of the Wiener process belong almost surely. We consider stochastic differential equations with no global growth condition on the non-linearities and we describe the topological support of the laws of trajectories of the solutions in every Polish subspace of continuous functions into which the Besov–Orlicz space B1/2Φ,∞(0,T;Rd) is embedded compactly.
Example of a non-standard extreme-value law
Autoři
Haydn, N.; Kupsa, M.
Rok
2015
Publikováno
Ergodic Theory and Dynamical Systems. 2015, 35(6), 1902-1912. ISSN 0143-3857.
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It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme-value laws which in the limit converge to one of the three standard distributions known for independently and identically distributed processes, namely Gumbel, Fréchet and Weibull distributions. In this short note, we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.