Mutual Singularity Research Articles

Mutual singularities of overlapping attractor and repeller.

We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker's map); two other examples, the Anosov-Möbius and the Chirikov-Möbius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored numerically. We demonstrate that although for these maps the stable and unstable directions are not orthogonal to each other, the relative Rényi and Kullback-Leibler dimensions as well as the mutual singularity spectra for the attractor and repeller can be well approximated under orthogonality assumption of two fractals.

On the mutual singularity of Hewitt–Stromberg measures for which the multifractal functions do not necessarily coincide

In this paper, we attain the multifractal Hewitt–Stromberg dimension functions of Moran measures associated with homogeneous Moran fractals and show that the multifractal Hewitt–Stromberg measures are mutually singular for which the multifractal functions do not necessarily coincide. In particular, we give a positive answer to questions posed in Attia and Selmi (J Geom Anal 31:825-862, 2021) and discuss some interesting examples.

Remarks on the mutual singularity of multifractal measures

In the present work, we study the mutual singularity of multifractal Hausdorff and packing measures which provide a positive answer to Olsen’s questions in a more general framework. Our main results apply to a family of measures supported by the full 5-adic grid of [0, 1], namely the quasi-Bernoulli measures.

The mutual singularity of the relative multifractal measures

Abstract M. Das proved that the relative multifractal measures are mutually singular for the self-similar measures satisfying the significantly weaker open set condition. The aim of this paper is to show that these measures are mutually singular in a more general framework. As examples, we apply our main results to quasi-Bernoulli measures.

Another example of the mutual singularity of multifractal measures

We propose an example for which the multifractal Hausdorff and packing measures are mutually singular.

The mutual singularity of multifractal measures for some non-regular Moran fractals

The mutual singularity of multifractal measures for some non-regular Moran fractals

Matrix measures and finite rank perturbations of self-adjoint operators

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new perturbation theoretic results. Our main result is a generalization of the Aronszajn–Donoghue theorem about the mutual singularity of the singular parts of the spectrum for rank one perturbations to the case of finite rank perturbations. Simple direct sum type examples indicate that an exact generalization is not possible. However, in this paper we introduce the notion of vector mutual singularity for the matrix-valued measures and show that if we use this notion, the mutual singularity still holds for the finite rank perturbations. As for the scalar spectral measures and the classical mutual singularity, we show that the singular parts are mutually singular for almost all perturbations. One of the ways to prove that is to use a generalization of the Aleksandrov’s spectral averaging to the matrix-valued measures, which is also one of the main results of this paper. Finally, the spectral representation of the perturbed operator is obtained. The matrix Muckenhoupt A_2 condition appears naturally there, and it plays an important role in establishing the vector mutual singularity of the spectral measures.

On the mutual singularity of multifractal measures

<p style='text-indent:20px;'>The aim of this article is to show that the multifractal Hausdorff and packing measures are mutually singular, which in particular provides an answer to Olsen's questions.</p>

The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One

Abstract Let $\Omega \subset{\mathbb{R}}^{n+1}$ be open and let $E\subset \partial \Omega $ with $0<{\mathcal{H}}^s(E)<\infty $, for some $s\in (n,n+1)$, satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure ${\mathcal{H}}^s$ on $E$. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that $\Omega $ is a uniform domain.

Spectral theory of Fourier–Stieltjes algebras

In this paper we start studying spectral properties of the Fourier-Stieltjes algebras, largely following Zafran's work on the algebra of measures on a locally compact group. We show that for a large class of discrete groups the Wiener-Pitt phenomenon occurs, i.e. the spectrum of an element of the Fourier-Stieltjes algebra is not captured by its range. We also investigate the notions of absolute continuity and mutual singularity in this setting; non-commutativity forces upon us two distinct versions of support of an element, indicating a crucial difference between this setup and the realm of Abelian groups. In spite of these difficulties, we also show that one can introduce and use generalised characters to prove a criterion on belonging of a multiplicative-linear functional to the Shilov boundary of the Fourier-Stieltjes algebra.

On the Use of Reproducing Kernel Hilbert Spaces in Functional Classification

ABSTRACTThe Hájek–Feldman dichotomy establishes that two Gaussian measures are either mutually absolutely continuous with respect to each other (and hence there is a Radon–Nikodym density for each measure with respect to the other one) or mutually singular. Unlike the case of finite-dimensional Gaussian measures, there are nontrivial examples of both situations when dealing with Gaussian stochastic processes. This article provides: (a) Explicit expressions for the optimal (Bayes) rule and the minimal classification error probability in several relevant problems of supervised binary classification of mutually absolutely continuous Gaussian processes. The approach relies on some classical results in the theory of reproducing kernel Hilbert spaces (RKHS). (b) An interpretation, in terms of mutual singularity, for the so-called “near perfect classification” phenomenon. We show that the asymptotically optimal rule proposed by these authors can be identified with the sequence of optimal rules for an approximating sequence of classification problems in the absolutely continuous case. (c) As an application, we discuss a natural variable selection method, which essentially consists of taking the original functional data X(t), t ∈ [0, 1] to a d-dimensional marginal (X(t1), …, X(td)), which is chosen to minimize the classification error of the corresponding Fisher’s linear rule. We give precise conditions under which this discrimination method achieves the minimal classification error of the original functional problem. Supplementary materials for this article are available online.

Optimal stopping with random maturity under nonlinear expectations

We analyze an optimal stopping problem supγ∈TE¯0[Yγ∧τ0] with random maturity τ0 under a nonlinear expectation E¯0[⋅]:=supP∈PEP[⋅], where P is a weakly compact set of mutually singular probabilities. The maturity τ0 is specified as the hitting time to level 0 of some continuous index process X at which the payoff process Y is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.We utilize a martingale approach to construct an optimal pair (P∗,γ∗) for sup(P,γ)∈P×TEP[Yγ∧τ0], in which γ∗ is the first time Y meets the limit Z of its approximating E¯−Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process Y, we approximate τ0 by an increasing sequence of Lipschitz continuous stopping times and approximate Y by a sequence of uniformly continuous processes.

On the robust Dynkin game

We analyze a robust version of the Dynkin game over a set $\mathcal{P}$ of mutually singular probabilities. We first prove that conservative player’s lower and upper value coincide (let us denote the value by $V$). Such a result connects the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$ up to the first time $\tau_{*}$ when $V$ meets the lower payoff process $L$. If the probability set $\mathcal{P}$ is weakly compact, one can even find an optimal triplet $(\mathbb{P}_{*},\tau_{*},\gamma_{*})$ for the value $V_{0}$. The mutual singularity of probabilities in $\mathcal{P}$ causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.

On the Robust Optimal Stopping Problem

We study a robust optimal stopping problem with respect to a set ${\cal P}$ of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its payoff while an adverse player wants to minimize this payoff by choosing an evaluation criteria from ${\cal P}$. We show that the upper Snell envelope $\overline{Z}$ of the reward process $Y$ is a supermartingale with respect to an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$, and $\overline{Z}$ is further an $\underline{\mathscr{E}}$-martingale up to the first time $\tau^*$ when $\overline{Z}$ meets $Y$. Consequently, $\tau^*$ is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circum...

Some remarks on [formula omitted]-adic Riesz products

We study the precise conditions for mutual absolute continuity and mutual singularity of two p-adic Riesz products, defined respectively by two coefficients sequences ak,bk. Our conditions are different from that appear in (Fan and Zhang, 2009) [6]. This paper can be viewed as a generalization of (Fan and Zhang, 2009) [6] in this topic. We also calculate the Hausdorff dimension of Riesz products.

On the Robust Optimal Stopping Problem

We study a robust optimal stopping problem with respect to a set P of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its pay-off while an adverse player wants to minimize this payoff by choosing an evaluation criteria from P. We show that the upper Snell envelope Z of the reward process Y is a supermartingale with respect to an appropriately defined nonlinear expectation E, and Z is further an E-martingale up to the first time tau when Z meets Y . Consequently, tau is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circumvent using some new methods.

Unsupervised classification and analysis of objects described by nonparametric probability distributions

AbstractVarious objects can be summarily described by probability distributions: groups of raw data, paths of stochastic processes, neighborhoods of an image pixel and so on. Dealing with nonparametric distributions, we propose a method for classifying such objects by estimating a finite mixture of Dirichlet distributions when the observed distributions are assumed to be outcomes of a finite mixture of Dirichlet processes. We prove the consistency of such a classification by using the mutual singularity of two distinct Dirichlet processes and the martingale convergence theorem. Moreover, this consistency allows us to use some standard data analysis and statistical methods for analyzing the class labels of these objects. © 2012 Wiley Periodicals, Inc. Statistical Analysis and Data Mining, 2012

Copulas and associated fractal sets

The problem of constructing copulas whose supports are fractals has been studied by Fredricks, Nelsen and Rodríguez-Lallena [G.A. Fredricks, R.B. Nelsen, J.A. Rodríguez-Lallena, Copulas with fractal supports, Insurance Math. Econom. 37 (1) (2005) 42–48]. In this paper we continue on the path traced by these authors. We provide different types of families of self-similar copulas using techniques from Probability and Ergodic Theory to give properties on subsets of their fractal supports. In particular, we give new examples for those copulas and we analyze related topics with mutual singularity of the associated measures, Hausdorff dimension, and the connectedness of their supports.

Random $p$-adic Riesz products: Continuity, singularity, and dimension

We study precise conditions for mutual absolute continuity and mutual singularity of two random p-adic Riesz products, defined respectively by two sequences of coefficients a k , b k . Our conditions and assertions are specific to the p-adic case. We also calculate explicitly the Hausdorff dimension, and in case the defining coefficients are constant, we have an integral representation of the dimension formula with a rapid convergence rate p -k .

Schrödinger representation for the polarized Gowdy model

The polarized T3 Gowdy model is, in a standard gauge, characterized by a point particle degree of freedom and a scalar field degree of freedom obeying a linear field equation on R × S1. The Fock representation of the scalar field has been well studied. We construct the Schrödinger representation for the scalar field at a fixed value of the Gowdy time in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure. We show that ‘typical’ field configurations are slightly more singular than square-integrable functions on the circle. For each time the corresponding Schrödinger representation is unitarily equivalent to the Fock representation, and hence all the Schrödinger representations are equivalent. However, the failure of unitary implementability of time evolution in this model manifests itself in the mutual singularity of the Gaussian measures at different times.