Group theoretic dimension of stationary symmetric \alpha-stable random fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.Comment: To appear in Journal of Theoretical Probability. Affiliation of the authors are update

A New Method to Estimate the Noise in Financial Correlation Matrices

Financial correlation matrices measure the unsystematic correlations between stocks. Such information is important for risk management. The correlation matrices are known to be ``noise dressed''. We develop a new and alternative method to estimate this noise. To this end, we simulate certain time series and random matrices which can model financial correlations. With our approach, different correlation structures buried under this noise can be detected. Moreover, we introduce a measure for the relation between noise and correlations. Our method is based on a power mapping which efficiently suppresses the noise. Neither further data processing nor additional input is needed.Comment: 25 pages, 8 figure

Stochastic stability versus localization in chaotic dynamical systems

We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the so called bounded variation approach combined with the ergodic theorem of Ionescu-Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of ``traps'' under the action of random perturbations.Comment: 27 pages, LaTe

Some results about diagonal operators on Köthe echelon spaces

[EN] Several questions about diagonal operators between Köthe echelon spaces are investigated: (1) The spectrum is characterized in terms of the Köthe matrices defining the spaces, (2) It is characterized when these operators are power bounded, mean ergodic or uniformly mean ergodic, and (3) A description of the topology in the space of diagonal operators induced by the strong topology on the space of all operators is given.This research was partially supported by MINECO Project MTM2016-76647-P and the grant PAID-01-16 of the Universitat Politècnica de València.Rodríguez-Arenas, A. (2019). Some results about diagonal operators on Köthe echelon spaces. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):2959-2968. https://doi.org/10.1007/s13398-019-00663-yS295929681134Agathen, S., Bierstedt, K.D., Bonet, J.: Projective limits of weighted (LB)-spaces of continuous functions. Arch. Math. 92, 384–398 (2009)Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34(2), 401–436 (2009)Bennett, G.: Some elementary inequalities. Quart. J. Math. 38, 401–425 (1987)Bennett, G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. (1996). https://doi.org/10.1090/memo/0576Bierstedt, K.D.: An introduction to locally convex inductive limits, Functional analysis and its applications (Nice, 1986), 35–133, ICPAM Lecture Notes. World Sci. Publishing, Singapore (1988)Bierstedt, K.D., Bonet, J.: Some aspects of the modern theory of Fréchet spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97(2), 159–188 (2003)Bierstedt, K.D., Meise, R., Summers, W.H.: Köthe sets and Köthe sequence spaces, Functional Analysis, Holomorphy and Approximation Theory. North-Holland Math. Studies 71, 27–91 (1982)Bonet, J., Jordá, E., Rodríguez-Arenas, A.: Mean ergodic multiplication operators on weighted spaces of continuous functions. Mediterr. J. Math 15, 108 (2018)Crofts, G.: Concerning perfect Fréchet spaces and transformations. Math. Ann. 182, 67–76 (1969)Kellogg, C.N.: An extension of the Hausdorff–Young theorem. Michig. Math. J. 18, 121–127 (1971)Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, New York (1997)Vasilescu, F.H.: Analytic Functional Calculus and Spectral Decompositions. D. Reidel Publ. Co., Dordrecht (1982)Wengenroth, J.: Derived Functors in Functional Analysis. Springer, Berlin (2003)Yosida, K.: Functional Analysis. Springer, Berlin (1980

A Pathwise Ergodic Theorem for Quantum Trajectories

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.Comment: 8 page

Infinite ergodic theory and Non-extensive entropies

We bring into account a series of result in the infinite ergodic theory that we believe that they are relevant to the theory of non-extensive entropie

Parametric estimation of the driving L\'evy process of multivariate CARMA processes from discrete observations

We consider the parametric estimation of the driving L\'evy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid $(0,h,2h,...)$. Beginning with a new state space representation, we develop a method to recover the driving L\'evy process exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery of unit increments of the driving L\'evy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen dependent on $N$, the length of the observation horizon, such that $N h_N$ converges to zero as $N$ tends to infinity, then any suitable generalized method of moments estimator based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on the true increments, and is, in particular, asymptotically normally distributed.Comment: 38 pages, four figures; to appear in Journal of Multivariate Analysi

New evidence for an early settlement of the Yucatán Peninsula, Mexico: The Chan Hol 3 woman and her meaning for the Peopling of the Americas.

Human presence on the Yucatán Peninsula reaches back to the Late Pleistocene. Osteological evidence comes from submerged caves and sinkholes (cenotes) near Tulum in the Mexican state of Quintana Roo. Here we report on a new skeleton discovered by us in the Chan Hol underwater cave, dating to a minimum age of 9.9±0.1 ky BP based on 230Th/U-dating of flowstone overlying and encrusting human phalanges. This is the third Paleoindian human skeleton with mesocephalic cranial characteristics documented by us in the cave, of which a male individual named Chan Hol 2 described recently is one of the oldest human skeletons found on the American continent. The new discovery emphasizes the importance of the Chan Hol cave and other systems in the Tulum area for understanding the early peopling of the Americas. The new individual, here named Chan Hol 3, is a woman of about 30 years of age with three cranial traumas. There is also evidence for a possible trepanomal bacterial disease that caused severe alteration of the posterior parietal and occipital bones of the cranium. This is the first time that the presence of such disease is reported in a Paleoindian skeleton in the Americas. All ten early skeletons found so far in the submerged caves from the Yucatán Peninsula have mesocephalic cranial morphology, different to the dolicocephalic morphology for Paleoindians from Central Mexico with equivalent dates. This supports the presence of two morphologically different Paleoindian populations for Mexico, coexisting in different geographical areas during the Late Pleistocene-Early Holocene

Prophet Inequalities for IID Random Variables from an Unknown Distribution

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, . . . , Xn drawn independently from a distribution F , the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we have E[Xτ ] ≥ α · E[maxt Xt ]. What makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X1, . . . , Xt and on the distribution F . For a long time the best known bound for the problem had been α ≥ 1 − 1/e ≈ 0.632, but quite recently a tight bound of α ≈ 0.745 was obtained. The case where F is unknown, such that the decision whether τ = t may depend only on the values of the random variables X1, . . . , Xt , is equally well motivated but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F , and show that even with o(n) samples α ≤ 1/e. On the other hand, n samples allow for a significant improvement, while O(n2) samples are equivalent to knowledge of the distribution: specifically, with n samples α ≥ 1 − 1/e ≈ 0.632 and α ≤ ln(2) ≈ 0.693, and with O(n2) samples α ≥ 0.745 − ε for any ε > 0