scholarly journals A formula for hidden regular variation behavior for symmetric stable distributions

Extremes ◽  
2020 ◽  
Vol 23 (4) ◽  
pp. 667-691
Author(s):  
Malin Palö Forsström ◽  
Jeffrey E. Steif
Keyword(s):  
Power Law ◽  
Spectral Measure ◽  
Regular Variation ◽  
Stable Distributions ◽  
Decay Rates ◽  
Random Vectors ◽  
Power Law Decay ◽  
Exact Power ◽  
Hidden Regular Variation ◽  
Stable Random Vectors

Abstract We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in “different directions”, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

scholarly journals Sparse regular variation

2021 ◽  
Vol 53 (4) ◽  
pp. 1115-1148
Author(s):  
Nicolas Meyer ◽  
Olivier Wintenberger
Keyword(s):  
Weak Convergence ◽  
Extreme Events ◽  
Spectral Measure ◽  
Regular Variation ◽  
Theoretical Framework ◽  
Efficient Algorithms ◽  
Dependence Structure ◽  
Random Vectors ◽  
Sparsity Structure ◽  
Natural Way

AbstractRegular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

Straight-sided solutions to classical and modified plume flux equations

2011 ◽  
Vol 680 ◽  
pp. 564-573 ◽  
Author(s):  
N. B. KAYE ◽  
M. M. SCASE
Keyword(s):  
Chemical Reactions ◽  
Power Law ◽  
Analytical Solutions ◽  
Classical Model ◽  
Decay Rates ◽  
Buoyancy Flux ◽  
Governing Equations ◽  
Jet Length ◽  
Source Radius ◽  
Power Law Decay

The classical plume model due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23) is re-cast in terms of the non-dimensional plume radius, the plume ‘laziness’ defined as the squared ratio of the source radius and the jet length, and the buoyancy flux. It is shown that many of the key results of this classical model can then be read straight from the equations without recourse to solving them. Based on this observation, derivative models that consider plumes propagating through stratified environments or undergoing chemical reactions are similarly re-cast. We show again that key results can be read straight from the governing equations and results that have previously only been demonstrated numerically can be found analytically. In particular, we unify two previously distinct models that consider plumes propagating through stable and unstable stratified environments whose stratification has a power-law dependence on height. We present analytical solutions for the range of stratification power-law decay rates for which straight-sided plumes are possible. This result unifies the sets of solutions by Batchelor (Q. J. R. Meteorol. Soc., vol. 80, 1954, pp. 339–358) and Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248). We are able to explain the unstable behaviour previously found when the power lies in the range (−4, −8/3). Finally we show that this method also has limited advantages when applied to plumes with unsteady source conditions.

scholarly journals Characterizations and Examples of Hidden Regular Variation

Extremes ◽  
2004 ◽  
Vol 7 (1) ◽  
pp. 31-67 ◽  
Author(s):  
Krishanu Maulik ◽  
Sidney Resnick
Keyword(s):  
Regular Variation ◽  
Hidden Regular Variation

scholarly journals When an oscillating center in an open system undergoes power law decay

2018 ◽  
Vol 57 (3) ◽  
pp. 750-768 ◽  
Author(s):  
Sandip Saha ◽  
Gautam Gangopadhyay
Keyword(s):  
Power Law ◽  
Open System ◽  
Power Law Decay

scholarly journals Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone

2018 ◽  
Vol 5 (5) ◽  
Author(s):  
Nils O. Abeling ◽  
Lorenzo Cevolani ◽  
Stefan Kehrein
Keyword(s):  
Correlation Function ◽  
Power Law ◽  
Light Cone ◽  
Relativistic Quantum ◽  
Initial State ◽  
Quantum Theories ◽  
Luttinger Model ◽  
Power Law Decay ◽  
Exactly Solvable ◽  
Exponentially Small

In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.

On dispersion of stable random vectors and its application in the prediction of multivariate stable processes

1994 ◽  
Vol 31 (3) ◽  
pp. 691-699 ◽  
Author(s):  
A. Reza Soltani ◽  
R. Moeanaddin
Keyword(s):  
Stable Process ◽  
Stable Processes ◽  
Random Vectors ◽  
Linear Predictor ◽  
Error Vector ◽  
Best Linear Predictor ◽  
Definition Of ◽  
Stable Random Vectors

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

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