Heavy-tailed Fluctuations Research Articles

Tournament rewards and heavy tails

Heavy-tailed fluctuations are common in many environments, such as sales of creative and innovative products or the financial sector. We study how the presence of heavy tails in the distribution of shocks affects the optimal allocation of prizes in rank-order tournaments à la Lazear and Rosen (1981). While a winner-take-all prize schedule maximizes aggregate effort for light-tailed shocks, prize sharing becomes optimal when shocks acquire heavy tails, increasingly so following a skewness order. Extreme prize sharing—rewarding all ranks but the very last—is optimal when shocks have a decreasing failure rate, such as power laws. Hence, under heavy-tailed uncertainty, typically associated with strong inequality in the distribution of gains, providing incentives and reducing inequality go hand in hand.

Macroscopic and edge behavior of a planar jellium

We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disk. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge-neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires a low enough temperature or high enough total smeared charge. This condition does not allow, at the same time, total charge-neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester–Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disk as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester–Krishnapur spherical ensemble. We also touch upon a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.

Multifractal detrended fluctuation analysis: Practical applications to financial time series

To analyze financial time series exhibiting volatility clustering or other highly irregular behavior, we exploit multifractal detrended fluctuation analysis (MF-DFA). We summarize the use of local Hölder exponents, generalized Hurst exponents, and the multifractal spectrum in characterizing the way that the sample paths of a multifractal stochastic process exhibit light- or heavy-tailed fluctuations as well as short- or long-range dependence on different time scales. We detail the development of a robust, computationally efficient software tool for estimating the multifractal spectrum from a time series using MF-DFA, with special emphasis on selecting the algorithm’s parameters. The software is tested on simulated sample paths of Brownian motion, fractional Brownian motion, and the binomial multiplicative process to verify the accuracy of the resulting multifractal spectrum estimates. We also perform an in-depth analysis of General Electric’s stock price using conventional time series models, and we contrast the results with those obtained using MF-DFA.

Heavy-Tailed Fluctuations in the Spiking Output Intensity of Semiconductor Lasers with Optical Feedback.

Although heavy-tailed fluctuations are ubiquitous in complex systems, a good understanding of the mechanisms that generate them is still lacking. Optical complex systems are ideal candidates for investigating heavy-tailed fluctuations, as they allow recording large datasets under controllable experimental conditions. A dynamical regime that has attracted a lot of attention over the years is the so-called low-frequency fluctuations (LFFs) of semiconductor lasers with optical feedback. In this regime, the laser output intensity is characterized by abrupt and apparently random dropouts. The statistical analysis of the inter-dropout-intervals (IDIs) has provided many useful insights into the underlying dynamics. However, the presence of large temporal fluctuations in the IDI sequence has not yet been investigated. Here, by applying fluctuation analysis we show that the experimental distribution of IDI fluctuations is heavy-tailed, and specifically, is well-modeled by a non-Gaussian stable distribution. We find a good qualitative agreement with simulations of the Lang-Kobayashi model. Moreover, we uncover a transition from a less-heavy-tailed state at low pump current to a more-heavy-tailed state at higher pump current. Our results indicate that fluctuation analysis can be a useful tool for investigating the output signals of complex optical systems; it can be used for detecting underlying regime shifts, for model validation and parameter estimation.

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamichysteresis

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes aGaussian character of the noise term describing interaction of the analyzed system with itscomplex surroundings. An additional assumption about the existence of timescaleseparation between the dynamics of the measured observable and the typical timescale ofthe noise allows external fluctuations to be modeled as temporally uncorrelated andtherefore white. However, in many natural phenomena the assumptions concerning theabove mentioned properties of ‘Gaussianity’ and ‘whiteness’ of the noise can beviolated. In this context, in contrast to the spatiotemporal coupling characterizinggeneral forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flightscorrespond to the class of Markov processes which can still be interpreted aswhite, but distributed according to a more general, infinitely divisible, stable andnon-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifestinteresting physical properties and have been addressed in various scenarios ofphysical transport exhibiting a superdiffusive behavior. Here we present a briefoverview of our recent investigations aimed at understanding features of stochasticdynamics under the influence of Lévy white noise perturbations. We find that thearchetypal phenomena of noise-induced ordering are robust and can be detected also insystems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinitevariance.