Sparse regular variation

Regular 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.

Aliasing Free For Mixed Spectra for Stable Processes

This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.

Quantitative Global-Local Mixing for Accessible Skew Products

We study global-local mixing for a family of accessible skew products with an exponentially mixing base and non-compact fibers, preserving an infinite measure. For a dense set of almost periodic global observables, we prove rapid mixing, and for a dense set of global observables vanishing at infinity, we prove polynomial mixing. More generally, we relate the speed of mixing to the “low frequency behavior” of the spectral measure associated to our global observables. Our strategy relies on a careful choice of the spaces of observables and on the study of a family of twisted transfer operators.

A formula for hidden regular variation behavior for symmetric stable distributions

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.

NUMBER THEORY PROBLEMS RELATED TO THE SPECTRUM OF CANTOR-TYPE MEASURES WITH CONSECUTIVE DIGITS

For integers $p,b\geq 2$ , let $D=\{0,1,\ldots ,b-1\}$ be a set of consecutive digits. It is known that the Cantor measure $\unicode[STIX]{x1D707}_{pb,D}$ generated by the iterated function system $\{(pb)^{-1}(x+d)\}_{x\in \mathbb{R},d\in D}$ is a spectral measure with spectrum $$\begin{eqnarray}\unicode[STIX]{x1D6EC}(pb,S)=\bigg\{\mathop{\sum }_{j=0}^{\text{finite}}(pb)^{j}s_{j}:s_{j}\in S\bigg\},\end{eqnarray}$$ where $S=pD$ . We give conditions on $\unicode[STIX]{x1D70F}\in \mathbb{Z}$ under which the scaling set $\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6EC}(pb,S)$ is also a spectrum of $\unicode[STIX]{x1D707}_{pb,D}$ . These investigations link number theory and spectral measures.