Search ResultsShowing 1-20 of 23
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arXiv:2309.12196 (Published 2023-09-21)
A variational approach to free probability
Comments: 29 pagesLet $\mu$ and $\nu$ be compactly supported probability measures on the real line with densities with respect to Lebesgue measure. We show that for all large real $z$, if $\mu \boxplus \nu$ is their additive free convolution, we have \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) = \sup_{\Pi} \left\{ \mathbb{E}_\Pi[\log(z - (X+Y)] - \mathcal{E}[\Pi]+\mathcal{E}[\mu]+\mathcal{E}[\nu] \right\}, \end{equation*} where the supremum is taken over all probability laws $\Pi$ on $\mathbb{R}^2$ for a pair of real-valued random variables $(X,Y)$ with respective marginal laws $\mu$ and $\nu$, and given a probability law $P$ with density function $f$ on $\mathbb{R}^k$, $\mathcal{E}[P] := \int_{\mathbb{R}^k} f \log f$ is its classical entropy. We prove similar formulas for the multiplicative free convolution $\mu \boxtimes \nu$ and the free compression $[\mu]_\tau$ of probability laws. The maximisers in our variational descriptions of these free operations on measures can be computed explicitly, and from these we can then deduce the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution. We use our formulation to derive several new inequalities relating free and classical convolutions of random variables, such as \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) \geq \mathbb{E}[\log(z - (X+Y)], \end{equation*} valid for all large $z$, where on the right-hand side $X,Y$ are independent classical random variables with respective laws $\mu,\nu$. Our approach is based on applying a large deviation principle on the symmetric group to the celebrated quadrature formulas of Marcus, Spielman and Srivastava.
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arXiv:2211.14857 (Published 2022-11-27)
Information Measures for Entropy and Symmetry
Entropy and information can be considered dual: entropy is a measure of the subspace defined by the information constraining the given ambient space. Negative entropies, arising in na\"ive extensions of the definition of entropy from discrete to continuous settings, are byproducts of the use of probabilities, which only work in the discrete case by a fortunate coincidence. We introduce notions such as sup-normalization and information measures, which allow for the appropriate generalization of the definition of entropy that keeps with the interpretation of entropy as a subspace volume. Applying this in the context of topological groups and Haar measures, we elucidate the relationship between entropy, symmetry, and uniformity.
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arXiv:2209.13634 (Published 2022-09-27)
Non-archimedean Schur representations of $\mathrm{GL}(n,R)$ and their invariant lattices
Comments: 14 pages, 1 FigureLet a $K$ be a non-archimedean discretely valued field and $R$ its valuation ring. Given a vector space $V$ of dimension $n$ over $K$ and a partition $\lambda$ of an integer $d$, we study the problem of determining the invariant lattices in the Schur module $S_\lambda(V)$ under the action of the group $\mathrm{GL}(n,R)$. When $K$ is a non-archimedean local field, our results determine the $\mathrm{GL}(n,R)$-invariant Gaussian distributions on $S_\lambda(V)$.
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arXiv:2208.07641 (Published 2022-08-16)
Higher order concentration on Stiefel and Grassmann manifolds
Categories: math.PRWe prove higher order concentration bounds for functions on Stiefel and Grassmann manifolds equipped with the uniform distribution. This partially extends previous work for functions on the unit sphere. Technically, our results are based on logarithmic Sobolev techniques for the uniform measures on the manifolds. Applications include Hanson--Wright type inequalities for Stiefel manifolds and concentration bounds for certain distance functions between subspaces of $\mathbb{R}^n$.
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arXiv:2207.08418 (Published 2022-07-18)
Moment Methods on compact groups: Weingarten calculus and its applications
Comments: 20 pages, contribution to the ICM 2022, sections Analysis and ProbabilityA fundamental property of compact groups and compact quantum groups is the existence and uniqueness of a left and right invariant probability -- the Haar measure. This is a natural playground for classical and quantum probability, provided it is possible to compute its moments. Weingarten calculus addresses this question in a systematic way. The purpose of this manuscript is to survey recent developments, describe some salient theoretical properties of Weingarten functions, as well as applications of this calculus to random matrix theory, quantum probability, and algebra, mathematical physics and operator algebras.
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arXiv:2109.14890 (Published 2021-09-30)
The Weingarten Calculus
Comments: 15 pages, 3 figuresJournal: NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY (2022) 69, no. 5DOI: 10.1090/noti2474Tags: journal articleThis is a short introduction to Weingarten Calculus. Weingarten Calculus is a method to compute the joint moments of matrix variables distributed according to the Haar measure of compact groups.
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arXiv:2010.03908 (Published 2020-10-08)
On semilinear SPDEs with nonlinearities with polynomial growth
Let $\mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $\mathcal{X}$, let $F:\X\rightaarrow\X$ be a (smooth enough) function and let $\{W(t)\}_{t\geq 0}$ be a $\mathcal{X}$-valued cylindrical Wiener process. For $\alpha\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2\alpha-1}:Q^{1-2\alpha}(\mathcal{X})\subseteq\mathcal{X}\rightarrow\mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \begin{gather} \left\{\begin{array}{ll} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^{\alpha}dW(t), & t>0;\\ X(0,x)=x\in \mathcal{X}, \end{array} \right. \end{gather} and in its associated transition semigroup \begin{align} P(t)\varphi(x):=E[\varphi(X(t,x))], \qquad \varphi\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}; \end{align} where $B_b(\mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $\mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(\mathcal{X},\nu)$, where $\nu$ is the unique invariant probability measure of \eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincar\'e inequalities and we study the maximal Sobolev regularity for the stationary equation \[\lambda u-N_2 u=f,\qquad \lambda>0,\ f\in L^2(\mathcal{X},\nu);\] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(\mathcal{X},\nu)$.
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arXiv:1706.06968 (Published 2017-06-21)
Exact Coupling of Random Walks on Polish Groups
Comments: 14 pagesCategories: math.PRExact coupling of random walks is studied. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. In the Abelian case, it is shown that a random walk $S$ with step-length distribution $\mu$ started at $0$ admits a successful exact coupling with a version $S^x$ started at $x$ iff there is $n$ with $\mu^{n} \wedge \mu^{n}(x+\cdot) \neq 0$. In particular, this paper solves a problem posed by H.~Thorisson on successful exact coupling of random walks on $\mathbb{R}$. It is also noted that the set of such $x$ for which a successful exact coupling can be constructed is a Borel measurable group. Lastly, the weaker notion of possible exact coupling and its relationship to successful exact coupling is studied.
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arXiv:1704.08333 (Published 2017-04-26)
Point-shifts of Point Processes on Topological Groups
Comments: 26 pages (22 main + 4 appendix), 2 figuresCategories: math.PRThis paper focuses on covariant point-shifts of point processes on topological groups, which map points of a point process to other points of the point process in a translation invariant way. Foliations and connected components generated by point-shifts are studied, and the cardinality classification of connected components, previously known on Euclidean space, is generalized to unimodular groups. An explicit counterexample is also given on a non-unimodular group. Isomodularity of a point-shift is defined and identified as a key component in generalizations of Mecke's invariance theorem in the unimodular and non-unimodular cases. Isomodularity is also the deciding factor of when the reciprocal and reverse of a point-map corresponding to a bijective point-shift are equal in distribution. Finally, sufficient conditions for separating points of a point process are given.
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arXiv:1704.05205 (Published 2017-04-18)
Distances between Random Orthogonal Matrices and Independent Normals
Comments: 42 pagesCategories: math.PRLet $\Gamma_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let $ Z_n$ be the $p\times q$ upper-left submatrix of $\Gamma_n,$ where $p=p_n$ and $q=q_n$ are two positive integers. Let $G_n$ be a $p\times q$ matrix whose $pq$ entries are independent standard normals. In this paper we consider the distance between $\sqrt{n} Z_n$ and $ G_n$ in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. We prove that each of the first three distances goes to zero as long as $pq/n$ goes to zero, and not so this rate is sharp in the sense that each distance does not go to zero if $(p, q)$ sits on the curve $pq=\sigma n$, where $\sigma$ is a constant. However, it is different for the Euclidean distance, which goes to zero provided $pq^2/n$ goes to zero, and not so if $(p,q)$ sits on the curve $pq^2=\sigma n.$ A previous work by Jiang \cite{Jiang06} shows that the total variation distance goes to zero if both $p/\sqrt{n}$ and $q/\sqrt{n}$ go to zero, and it is not true provided $p=c\sqrt{n}$ and $q=d\sqrt{n}$ with $c$ and $d$ being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as $pq/n\to 0$ and the distance does not go to zero if $pq=\sigma n$ for some constant $\sigma$.
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arXiv:1601.04088 (Published 2016-01-14)
Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$
Comments: 12 pagesSubjects: 28C10We present the proof of Kolmogorov's strong law of large numbers in particular case and consider its applications for calculations of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$\beta$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a bigger set $D \subset(0,1)^{\infty}$ of uniformly distributed(in $(0,1)$) sequences strictly containing the set of all sequences of real numbers which can be presented in a form $(\{\alpha n\})_{n \in {\bf N}}$ for some irrational numbers $\alpha$ and for which $\ell_1^{\infty}(D)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.
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arXiv:1601.01931 (Published 2016-01-08)
Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions
Comments: 14pConsider the space $C$ of conjugacy classes of a unitary group $U(n+m)$ with respect to a smaller unitary group $U(m)$. It is known that for any element of the space $C$ we can assign canonically a matrix-valued rational function on the Riemann sphere (a Livshits characteristic function). In the paper we write an explicit expression for the natural measure on $C$ obtained as the pushforward of the Haar measure of the group $U(n+m)$ in the terms of characteristic functions.
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arXiv:1509.09261 (Published 2015-09-30)
Polar decomposition of scale-homogeneous measures with application to Lévy measures of strictly stable laws
Comments: 22 pagesCategories: math.PRA scaling on some space is a measurable action of the group of positive real numbers. A measure on a measurable space equipped with a scaling is said to be $\alpha$-homogeneous for some nonzero real number $\alpha$ if the mass of any measurable set scaled by any factor $t > 0$ is the multiple $t^{-\alpha}$ of the set's original mass. It is shown rather generally that given an $\alpha$-homogeneous measure on a measurable space there is a measurable bijection between the space and the Cartesian product of a subset of the space and the positive real numbers (that is, a "system of polar coordinates") such that the push-forward of the $\alpha$-homogeneous measure by this bijection is the product of a probability measure on the first component (that is, on the "angular" component) and an $\alpha$-homogeneous measure on the positive half-line (that is, on the "radial" component). This result is applied to the intensity measures of Poisson processes that arise in L\'evy-Khinchin-It\^o-like representations of infinitely divisible random elements. It is established that if a strictly stable random element in a convex cone admits a series representation as the sum of points of a Poisson process, then it necessarily has a LePage representation as the sum of i.i.d. random elements of the cone scaled by the successive points of an independent unit intensity Poisson process on the positive half-line each raised to the power $-\frac{1}{\alpha}$.
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arXiv:1507.01605 (Published 2015-07-06)
Leading Digit Laws on Linear Lie Groups
Comments: Version 1.0, 17 pages, 1 figureWe determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left invariance of Haar measure, we determine the power laws in significands from one matrix component of various such $G$. We also determine the leading digit distribution of a fixed number of components of a unit sphere, and find periodic behavior when the dimension of the sphere tends to infinity in a certain progression.
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arXiv:1506.04735 (Published 2015-06-15)
Infinite-Dimensional Monte-Carlo Integration
Comments: 9 pagesBy using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in $R^{\infty}$ described in [{G.R. Pantsulaia,} {\em On uniformly distributed sequences of an increasing family of finite sets in infinite-dimensional rectangles,} {Real Anal. Exchange.} {\bf 36 (2)} (2010/2011), 325--340 ], an infinite-dimensional Monte-Carlo integration is elaborated and the validity of some new Strong Law type theorems are obtained in this paper.
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arXiv:1506.02935 (Published 2015-05-18)
On Moore-Yamasaki-Kharazishvili type measures and the infinite powers of Borel diffused probability measures on ${\bf R}
Comments: 12pages. arXiv admin note: text overlap with arXiv:1502.07463The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].
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arXiv:1503.00865 (Published 2015-03-03)
Dimensions of graphs of prevalent continuous maps
Comments: 14 pages. arXiv admin note: text overlap with arXiv:1408.2176Let $K$ be an uncountable compact metric space and let $C(K,\mathbb{R}^d)$ denote the set of continuous maps $f\colon K \to \mathbb{R}^d$ endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$. As the main result of the paper we show that if $K$ has finitely many isolated points then the lower and upper box dimension of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ are $\underline{\dim}_B K+d$ and $\overline{\dim}_B K+d$, respectively. This generalizes a theorem of Gruslys, Jonu\v{s}as, Mijovi\`c, Ng, Olsen, and Petrykiewicz. We also prove that the packing dimension of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ is $\dim_P K+d$, generalizing a result of Balka, Darji, and Elekes. Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of the prevalent $f\in C(K,\mathbb{R}^d)$ equals $\dim_H K+d$. We give a simpler proof for this statement based on a method of Fraser and Hyde.
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arXiv:1411.5265 (Published 2014-11-19)
Généricité au sens probabiliste dans les difféomorphismes du cercle
Comments: 69 pages, in French, 49 figures. Preliminary version, to be published in Ensaios Matem\'aticos 27, Soc. Brasil. Mat. (2014)What kind of dynamics do we observe in general on the circle? It depends somehow on the interpretation of "in general". Everything is quite well understood in the topological (Baire) setting, but what about the probabilistic sense? The main problem is that on an infinite dimensional group there is no analogue of the Lebesgue measure, in a strict sense. There are however some analogues, quite natural and easy to define: the Malliavin-Shavgulidze measures provide an example and constitute the main character of this text. The first results show that there is no actual disagreement of general features of the dynamics in the topological and probabilistic frames: it is the realm of hyperbolicity! The most interesting questions remain however unanswered... This work, coming out from the author's Ph.D. thesis, constitutes an opportunity to review interesting results in mathematical topics that could interact more often: stochastic processes and one-dimensional dynamics. After an introductory overview, the following three chapters are a pedagogical summary of classical results about measure theory on topological groups, Brownian Motion, theory of circle diffeomorphisms. Then we present the construction of the Malliavin-Shavgulidze measures on the space of interval and circle $C^1$ diffeomorphisms, and discuss their key property of quasi-invariance. The last chapter is devoted to the study of dynamical features of a random Malliavin-Shavgulidze diffeomorphism.
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Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps
Comments: 42 pagesThe notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let $K$ be an uncountable compact metric space. We prove that the prevalent $f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of $y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open set having full measure with respect to the occupation measure $\lambda^m \circ f^{-1}$, where $\dim_H$ and $\lambda^m$ denote the Hausdorff dimension and the $m$-dimensional Lebesgue measure, respectively. We also prove an analogous result when $[0,1]^m$ is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions $f\in C[0,1]$ for which positively many level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we generalize a theorem of Antunovi\'c, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions $f\in C[0,1]$ for which $\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in $C[0,1]$. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde.
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Stationary Measures for Stochastic Differential Equations with Jumps
Comments: 13 pagesCategories: math.PRIn the paper, stationary measures of stochastic differential equations with jumps are considered. Under some general conditions, existence of stationary measures is proved through Markov measures and Lyapunov functions. Moreover, for two special cases, stationary measures are given by solutions of Fokker-Planck equations and long time limits for the distributions of system states.