Modeling Financial Data with Stable Distributions

Nolan, John P.

doi:10.1016/b978-044450896-6.50005-4

Cited by 139 publications

(172 citation statements)

References 28 publications

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“…Indeed, according to the developments in [45] and [46], and restricting our attention to the case 2 < α 3, we find the following result. Let p(∆t) behave as p(∆t) = λ∆t −α for large ∆t and let µ be its mean value.…”

Section: VI Discussion and Potential Relevance For Safety Sciencementioning

confidence: 94%

Statistical fluctuations in pedestrian evacuation times and the effect of social contagion

Nicolas¹,

Bouzat²,

Kuperman³

2016

Phys. Rev. E

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Mathematical models of pedestrian evacuation and the associated simulation software have become essential tools for the assessment of the safety of public facilities and buildings. While a variety of models are now available, their calibration and test against empirical data are generally restricted to global, averaged quantities; the statistics compiled from the time series of individual escapes ("microscopic" statistics) measured in recent experiments are thus overlooked. In the same spirit, much research has primarily focused on the average global evacuation time, whereas the whole distribution of evacuation times over some set of realizations should matter. In the present paper we propose and discuss the validity of a simple relation between this distribution and the "microscopic" statistics, which is theoretically valid in the absence of correlations. To this purpose, we develop a minimal cellular automaton, with novel features that afford a semi-quantitative reproduction of the experimental "microscopic" statistics. We then introduce a process of social contagion of impatient behavior in the model and show that the simple relation under test may dramatically fail at high contagion strengths, the latter being responsible for the emergence of strong correlations in the system. We conclude with comments on the potential practical relevance for safety science of calculations based on "microscopic" statistics.

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Section: VI Discussion and Potential Relevance For Safety Sciencementioning

confidence: 94%

Statistical fluctuations in pedestrian evacuation times and the effect of social contagion

Nicolas¹,

Bouzat²,

Kuperman³

2016

Phys. Rev. E

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“…Some special cases are symmetric, such as the Gaussian distribution, which corresponds to the limit α = 2, but a large skewness is generally seen, e.g., with the Landau distribution (α = 1), used by particle physicists (Groom & Klein 2000) to describe the distribution of the energy loss of a charged particle passing through matter. In particular, for 0 < α < 2 the PDF of an α-stable distribution asymptotically follows a power law of index 1 + α (Zaliapin et al 2005;Nolan 2012).…”

Section: Pareto Distributions and The Central Limit Theoremmentioning

confidence: 99%

The minijets-in-a-jet statistical model and the rms-flux correlation

Biteau

Giebels

2012

A&A

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The flux variability of blazars at very high energies does not have a clear origin. Flux variations on time scales down to the minute suggest that variability originates in the jet, where a relativistic boost can shorten the observed time scale, while the linear relation between the flux and its rms or the skewness of the flux distribution suggests that the variability stems from multiplicative processes, which are associated in some models with the accretion disk. We study the rms-flux relation and emphasize its link to Pareto distributions, characterized by a power-law probability density function. Such distributions are naturally generated within a minijets-in-a-jet statistical model, in which boosted emitting regions are isotropically oriented within the bulk relativistic flow of a jet. We prove that, within this model, the flux of a single minijet is proportional to its rms. This relation still holds when considering a large number of emitting regions, for which the distribution of the total flux is skewed and could be interpreted as being log-normal. The minijets-ina-jet statistical model reconciles the fast variations and the statistical properties of the flux of blazars at very high energies.

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“…The study of these stable distributions was begun by Paul Lévy in the 1920's. For recent results see [8].…”

Section: Lévy Distributed Sourcementioning

confidence: 99%

“…According to the generalized Central Limit Theorem, if particle production is a result of a multifold probabilistic process such as the branching of gluons into gluons into gluons etc., then the class of possible limiting probability distributions coincides with the class of Lévy distributions [8]. The characteristic function of symmetric stable distributions is [9] f (Q) = exp iQδ − |RQ|…”

Section: Lévy Distributed Sourcementioning

confidence: 99%