We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order are considered as well. We are interested in bounds with magic factors, i.e. roughly speaking $n$ also appears in the denominator. Special emphasis is given to the approximation by the $n$-fold convolution of the arithmetic mean of the distributions under consideration. As an application, we consider the multinomial approximation of the generalized multinomial distribution. It turns out that here the order of some bounds given in Roos (2001) and Loh (1992) can significantly be improved. In particular, it follows that a dimension factor can be dropped. Moreover, better accuracy is achieved in the context of symmetric distributions with finite support. In the course of proof, we use a basic Banach algebra technique for measures on a measurable Abelian group. Though this method was already used by Le Cam (1960), our central arguments seem to be new. We also derive new smoothness bounds for convolutions of probability distributions, which might be of independent interest.
The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions may be transferred to the estimation of the closeness of convolutions of probability distributions on convex polyhedra.
This expository paper is intended for a short self-contained introduction to the theory of infinite convolutions of probability measures on Polish semigroups. We give the proofs of the Rees decomposition theorem of completely simple semigroups, the Ellis--Żelazko theorem, the convolution factorization theorem of convolution idempotents, and the convolution factorization theorem of cluster points of infinite convolutions.
By using Lyapunov conditions, weak Poincaré inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.
We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.
Let $μ$ and $ν$ be two probability measures on $\R^d$, where $μ(\d x)= \e^{-V(x)}\d x$ for some $V\in C^1(\R^d)$. Explicit sufficient conditions on $V$ and $ν$ are presented such that $μ*ν$ satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, the recent results on the log-Sobolev inequality derived in \cite{Z} for convolutions of the Gaussian measure and compactly supported probability measures are improved and extended.
Based on the~method of subordinating functions we prove bounds for the minimal error of approximations of $n$-fold convolutions of probability measures by free infinitely divisible probability measures.
Our goal is to find classes of convolution semigroups on Lie groups $G$ that give rise to interesting processes in symmetric spaces $G/K$. The $K$-bi-invariant convolution semigroups are a well-studied example. An appealing direction for the next step is to generalise to right $K$-invariant convolution semigroups, but recent work of Liao has shown that these are in one-to-one correspondence with $K$-bi-invariant convolution semigroups. We investigate a weaker notion of right $K$-invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use genera…
The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of…
We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type $\int_0^t F_γ(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a Lévy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish crit…
In this paper we study the behaviour of convolution powers of probability measures $μ$ on $\Z$, such that $(μ(n))_{n\in \N}$ is completely monotone or such that $ν$ is centered with a second moment. In particular we exhibit many new examples of probability measures on $\Z$ having the so called Ritt property and whose convolution powers satisfy weak type maximal inequalities in $\ell^1(\Z)$.
Suppose Vν is the pseudo-variance function of the Cauchy-Stieltjes Kernel (CSK) family K+(ν) generated by a non degenerate probability measure ν with support bounded from above. We determine the formula for pseudo-variance function (or variance function Vν in case of existence) under boolean additive convolution power. This formulas is used to identify the relation between variance functions under Boolean Bercovici-Bata Bijection between probability measures. We also gives the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measure…
Let $μ$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $μ^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is to give an elementary description of $μ^{\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.
We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} ξ= \sum_{k=1}^\infty \frac{(-1)^{k+1}ξ_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{ξ_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $ξ$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_ξ)=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. …
Characterization problems in free probability are studied here. Using subordination of free additive and free multiplicative convolutions we generalize some known characterizations in free probability to random variables with unbounded support. Using this technique we also prove a new characterization of distributions of free random variables. A similar technique is used to study Laha-Lukacs regressions for monotonically independent random variables.
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convol…
We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree $\mathcal{T}$ of the $N$-regular tree (with vertices labeled by alternating strings), we define the convolution $\boxplus_{\mathcal{T}}(μ_1,\dots,μ_N)$ for arbitrary probability measures $μ_1$, ..., $μ_N$ on $\mathbb{R}$ using a certain fixed-poin…
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the empirical measure obtained from $n$ independent samples from $μ$ approaches $μ$ in the Wasserstein distance of any order. We prove sharp asymptotic and finite-sample results for this rate of convergence for general measures on general compact metric spaces. Our finite-sample results show the existence of multi-scale behavior, where measures can exhibit radic…
We generalize the Beckner's type Poincaré inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $μ\starν$, where $μ$ is the reference Gaussian measure and $ν$ is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and in…
In this paper we consider some generalizations of the classical d-dimensional Brownian risk model. This contribution derives some non-asymptotic bounds for simultaneous ruin probabilities of interest. In addition, we obtain non-asymptotic bounds also for the case of general trend functions and convolutions of our original risk model.
The superconvergence phenomenon is shown for products of free, identically distributed random variables. We also show that a certain Holder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free additive and multiplicative convolutions with arbitrary freely infinitely divisible laws and to free convolution semigroups.
We study the class $\mathcal{M}_{\mathrm{ratio}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the monotone convolution. We prove a sufficient condition that a rational function is the free $R$-transform of a probability distribution. Several examples are provided, including that of free deconvolution.
Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $μ$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, μ)$. In smooth ergodic theory there are various natural choices of $μ$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We sh…
Given a probability measure $μ$ over ${\mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stochastic localization argument to prove a general decomposition result of this type. In Eldan's theorem, the `number of components' is characterized by the entropy of the mixture, and `closeness to product' is characterized by the covariance matrix of each component. We present an elementary proof of Eldan's theorem which makes use of an information t…
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all …
The index Whittaker convolution operator, recently introduced by the authors, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. In this paper, we introduce the class of Lévy processes with respect to the index Whittaker convolution and study their basic properties. We prove that the square root of the Shiryaev process belongs to our family of Lévy process, and this is shown to yield a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demons…
In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions…
Numerical computation of convolutions in free probability theory
School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）；Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气与计算机工程系）
The local limit theorem describes the behavior of the convolution powers of a probability distribution supported on Z. In this work, we explore the role played by positivity in this classical result and study the convolution powers of the general class of complex valued functions finitely supported on Z. This is discussed as de Forest's problem in the literature and was studied by Schoenberg and Greville. Extending these earlier works and using techniques of Fourier analysis, we establish asymptotic bounds for the sup-norm of the convolution powers and prove extended local limit theorems perta…
In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…
Semigroups related to additive and multiplicative, free and Boolean convolutions
Belinschi和Nica在概率测度集上引入了一个复合半群。利用这个半群，他们引入了一个自由整除指标，从这个指标可以知道一个概率测度是否是自由无限整除的。在本文中，我们进一步研究这个指标，介绍了一个乘法版本，它能够显示许多属性。具体来说，在上半年的文件，我们计算如何变化的指标相对于自由和布尔幂权力，我们证明了自由和布尔1/2稳定的法律有自由整除指标等于无穷大，我们推导出一个上界的指标的Jacobi参数。这个上界只能通过自由Meixner分布实现。我们还证明了Bozejko的猜想说，布尔幂的概率测度亩0 < t < 1是自由无限整除，如果亩是这样。在本文的另一半，我们引入了一个类似的复合半群乘法卷积和定义自由整除指标，这些卷积。证明了单位圆上的一个概率测度关于乘法自由卷积是自由无限可分的当且仅当指标不小于1.我们还证明了乘性整除指标在自由幂和布尔幂幂下的变化，以及Bozejko猜想的乘性模拟。我们包括一个附录，其中的柯西分布和点的措施被证明是唯一的不动点的布尔自由Bercovici-Pata双射。
