Central limit theorems for correlated variables: some critical remarks

Hilhorst, H. J.

doi:10.1590/s0103-97332009000400005

Cited by 37 publications

(45 citation statements)

References 41 publications

(63 reference statements)

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“…Hilhorst has introduced [90] two interesting and paradigmatic examples which illustrate the difficulty. One of them is presented in his article [91] in this same volume. We shall here discuss his other example, which we present in what follows.…”

Section: Central Limit Theorems and Q-fourier Transformsmentioning

confidence: 99%

Nonadditive entropy and nonextensive statistical mechanics -an overview after 20 years

Tsallis¹

2009

Braz. J. Phys.

250

213

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Statistical mechanics constitutes one of the pillars of contemporary physics. Recognized as such -together with mechanics (classical, quantum, relativistic), electromagnetism and thermodynamics -, it is one of the mandatory theories studied at virtually all the intermediate-and advanced-level courses of physics around the world. As it normally happens with such basic scientific paradigms, it is placed at a crossroads of various other branches of knowledge. In the case of statistical mechanics, the standard theory -hereafter referred to as the Boltzmann-Gibbs (BG) statistical mechanics -exhibits highly relevant connections at a variety of microscopic, mesoscopic and macroscopic physical levels, as well as with the theory of probabilities (in particular, with the Central Limit Theorem, CLT ). In many circumstances, the ubiquitous efects of the CLT , with its Gaussian attractors (in the space of the distributions of probabilities), are present. Within this complex ongoing frame, a possible generalization of the BG theory was advanced in 1988 (C.T., J. Stat. Phys. 52, 479). The extension of the standard concepts is intended to be useful in those "pathological", and nevertheless very frequent, cases where the basic assumptions (molecular chaos hypothesis, ergodicity) for applicability of the BG theory would be violated. Such appears to be, for instance, the case in classical long-range-interacting many-body Hamiltonian systems (at the so-called quasi-stationary state). Indeed, in such systems, the maximal Lyapunov exponent vanishes in the thermodynamic limit N → ∞. This fact creates a quite novel situation with regard to typical BG systems, which generically have a positive value for this central nonlinear dynamical quantity. This peculiarity has sensible effects at all physical micro-, meso-and macroscopic levels. It even poses deep challenges at the level of the CLT . In the present occasion, after 20 years of the 1988 proposal, we undertake here an overview of some selected successes of the approach, and of some interesting points that still remain as open questions. Various theoretical, experimental, observational and computational aspects will be addressed.

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Section: Central Limit Theorems and Q-fourier Transformsmentioning

confidence: 99%

Nonadditive entropy and nonextensive statistical mechanics -an overview after 20 years

Tsallis¹

2009

Braz. J. Phys.

250

213

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“…The dynamics and limit distributions of constrained LFs are not well understood, except for processes subjected to long waiting times or in external potentials, mainly [8,31]. Several limit theorems also exist for specific problems of sums of correlated random variables [32], and a few random walks with infinite memory of their previous displacements have exactly solvable first moments [33][34][35]. Yet, very little is known on LFs composed of non-independent steps, in particular processes with self-attraction.…”

Section: Introductionmentioning

confidence: 99%

Slow Lévy flights

Boyer

Pineda

2016

Phys. Rev. E

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Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-thanBrownian dynamics. Here we show that Lévy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

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“…It is not so in general. Indeed, it has been shown by Hilhorst [Hilhorst, 2009[Hilhorst, , 2010 that, for a given value of q, one-parameter families of functions {f (x)} exist such that their q-FT does not depend on that parameter. Therefore the q-FT has not always an unique pre-image, i.e., although the q-FT is invertible within the closed class of q-Gaussians, it is not invertible in general.…”

Section: Q-fourier Transform and Discussion Of Its Inversementioning

confidence: 99%

“…Let us incidentally mention that the lack of inverse of the q-FT has made Hilhorst [Hilhorst, 2009[Hilhorst, , 2010] to disregard q-Gaussians as attractors. A detailed reply to his claim has been recently made available in , which reinforces that q-Gaussians are, in many respects, very special distributions.…”

Section: Q-generalized Central Limit Theoremsmentioning

confidence: 99%