Kolmogorov Distance Research Articles

Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

Distances Between Distributions Via Stein’s Method

We build on the formalism developed in Ernst et al. (First order covariance inequalities via Stein’s method, 2019) to propose new representations of solutions to Stein equations. We provide new uniform and nonuniform bounds on these solutions (a.k.a. Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, total variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive.

On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Vietoris-Rips and Čech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.

Poisson approximation with applications to stochastic geometry

This article compares the distributions of integer-valued random variables and Poisson random variables. It considers the total variation and the Wasserstein distance and provides, in particular, explicit bounds on the pointwise difference between the cumulative distribution functions. Special attention is dedicated to estimating the difference when the cumulative distribution functions are evaluated at 0. This permits to approximate the minimum (or maximum) of a collection of random variables by a suitable random variable in the Kolmogorov distance. The main theoretical results are obtained by combining the Chen-Stein method with size-bias coupling and a generalization of size-bias coupling for integer-valued random variables developed herein. A wide variety of applications are then discussed with a focus on stochastic geometry. In particular, transforms of the minimal circumscribed radius and the maximal inradius of Poisson-Voronoi tessellations as well as the minimal inter-point distance of the points of a Poisson process are considered and bounds for their Kolmogorov distances to extreme value distributions are derived.

Coordinated planning of multi-energy systems considering demand side response

Integrated multi-energy systems, due to their flexibility and complementary characteristics, are considered an important energy system structure for accommodating high renewable penetration, and thus have attracted great attention of global research recently. Based on the energy hub, a two-stage optimal design and planning method for regional integrated energy systems (IES) is proposed in this paper, Furthermore, the demand side response (DSR) is taken into consideration. The EnergyPlus simulation software is used to produce multi-energy load data for various types of buildings in urban areas. Then, the improved k-means data clustering method and the Kolmogorov Distance scene reduction technology are applied to generate three types of typical scenarios. The objective of optimal IES planning is to minimize total capital cost, O&M cost, and the shifting load compensation cost while making sure the system operates reliably and safely. The column-and-constraint (CCG) generation algorithm is implemented to achieve fast optimization solution. The simulation results show the effectiveness of the proposed method, and the benefits of IES with complementary multi-energies are reflected

Rate of convergence for traditional Pólya urns

AbstractConsider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.

On the error bound in the normal approximation for Jack measures

In this paper, we obtain uniform and non-uniform bounds on the Kolmogorov distance in the normal approximation for Jack deformations of the character ratio, by using Stein’s method and zero-bias couplings. Our uniform bound comes very close to that conjectured by Fulman (J. Combin. Theory Ser. A 108 (2004) 275–296). As a by-product of the proof of the non-uniform bound, we obtain a Rosenthal-type inequality for zero-bias couplings.

Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

Optimal Berry-Esséen bound for maximum likelihood estimation of the drift parameter in $$\alpha $$-Brownian bridge

Let $$T>0,\alpha >\frac{1}{2}$$ . In the present paper we consider the $$\alpha $$ -Brownian bridge defined as $$dX_t=-\alpha \frac{X_t}{T-t}dt+dW_t,\, 0\le t< T$$ , where W is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $$ \alpha $$ based on the continuous observation $$\{X_s,0\le s\le t\}$$ as $$t\uparrow T$$ . We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $$\frac{1}{\sqrt{\vert \log (T-t)\vert }}$$ , as $$t\uparrow T$$ . First we compute an upper bound and then find a lower bound with the same speed using Corollary 1 and Corollary 2 of Kim et al. (J Multivar Anal 155:284–304, 2017b) respectively.

Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent 23, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [8].We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus

In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents in the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in (possibly coloured) permutations, number of prime divisors (possibly within different residue classes) of a random integer, number of irreducible factors of a random polynomial, etc. One advantage of the approach developed in this paper is that it allows us to deal with approximations in higher dimensions as well. In this setting, we can explicitly see the influence of the correlations between the components of the random vectors in our asymptotic formulas.

Remark on rates of convergence to extreme value distributions via the Stein equations

Consider the maximum of independent and identically distributed random variables. The classical result says that the renormalized sample maximum converges to an extreme value distributions, under certain conditions on the distribution function. In the present paper, we shall study the uniform rate of the convergence with respect to the Kolmogorov distance in the framework of the Stein equations. Some typical examples are raised in the paper.

A Berry–Esseen Bound in the Smoluchowski–Kramers Approximation

In this paper, we use the Kolmogorov distance to investigate the Smoluchowski–Kramers approximation for stochastic differential equations. We obtain an explicit Berry–Esseen error bound for the rate of convergence. Our main tools are the techniques of Malliavin calculus.

Bootstrapping max statistics in high dimensions: Near-parametric rates under weak variance decay and application to functional and multinomial data

In recent years, bootstrap methods have drawn attention for their ability to approximate the laws of “max statistics” in high-dimensional problems. A leading example of such a statistic is the coordinatewise maximum of a sample average of $n$ random vectors in $\mathbb{R}^{p}$. Existing results for this statistic show that the bootstrap can work when $n\ll p$, and rates of approximation (in Kolmogorov distance) have been obtained with only logarithmic dependence in $p$. Nevertheless, one of the challenging aspects of this setting is that established rates tend to scale like $n^{-1/6}$ as a function of $n$. The main purpose of this paper is to demonstrate that improvement in rate is possible when extra model structure is available. Specifically, we show that if the coordinatewise variances of the observations exhibit decay, then a nearly $n^{-1/2}$ rate can be achieved, independent of $p$. Furthermore, a surprising aspect of this dimension-free rate is that it holds even when the decay is very weak. Lastly, we provide examples showing how these ideas can be applied to inference problems dealing with functional and multinomial data.

Convergence of eigenvector empirical spectral distribution of sample covariance matrices

The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Mar\v{c}enko-Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma^{1/2} X X^* \Sigma^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite matrix. We prove that the Kolmogorov distance between the expected VESD and the deformed MP distribution is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge \tau$ for some constant $\tau>0$. This result improves the previous one obtained in \cite{XYZ2013}, which gave the convergence rate $O(N^{-1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment, $\Sigma=I$ and $M<N$. Moreover, we also prove that under the finite $8$th moment assumption, the convergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for any fixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in \cite{XYZ2013}.

A bound on the rate of convergence in the central limit theorem for renewal processes under second moment conditions

AbstractA famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. We provide an explicit non-uniform bound for the renewal central limit theorem based on Stein’s method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry–Esseen theorem under second moment conditions.

Normal approximation for weighted sums under a second-order correlation condition

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.

Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem

Given a sequence $\{X_{n}\}_{n\geq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\stackrel{a.s.}{\longrightarrow }Y$ for a suitable random variable $Y:\Omega \rightarrow [0,1]$ satisfying $\mathsf{P}[X_{1}=x_{1},\dots ,X_{n}=x_{n}|Y]=Y^{\sum_{i=1}^{n}x_{i}}(1-Y)^{n-\sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $\frac{1}{n}\sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{\alpha }$ can be obtained for any index $\alpha \in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.

A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages

In this paper we obtain Berry–Esseén bounds on partial sums of functionals of heavy-tailed moving averages, including the linear fractional stable noise, stable fractional ARIMA processes and stable Ornstein–Uhlenbeck processes. Our rates are obtained for the Wasserstein and Kolmogorov distances, and depend strongly on the interplay between the memory of the process, which is controlled by a parameter $\alpha $, and its tail-index, which is controlled by a parameter $\beta $. In fact, we obtain the classical $1/\sqrt {n}$ rate of convergence when the tails are not too heavy and the memory is not too strong, more precisely, when $\alpha \beta >3$ or $\alpha \beta >4$ in the case of Wasserstein and Kolmogorov distance, respectively. Our quantitative bounds rely on a new second-order Poincaré inequality on the Poisson space, which we derive through a combination of Stein’s method and Malliavin calculus. This inequality improves and generalizes a result by Last, Peccati, Schulte [Probab. Theory Relat. Fields 165 (2016)].

The Lanczos Algorithm Under Few Iterations: Concentration and Location of the Output

We study the Lanczos algorithm where the initial vector is sampled uniformly from $\mathbb{S}^{n-1}$. Let $A$ be an $n \times n$ Hermitian matrix. We show that when run for few iterations, the output of Lanczos on $A$ is almost deterministic. More precisely, we show that for any $ \varepsilon \in (0, 1)$ there exists $c >0$ depending only on $\varepsilon$ and a certain global property of the spectrum of $A$ (in particular, not depending on $n$) such that when Lanczos is run for at most $c \log n$ iterations, the output Jacobi coefficients deviate from their medians by $t$ with probability at most $\exp(-n^\varepsilon t^2)$ for $t<\Vert A \Vert$. We directly obtain a similar result for the Ritz values and vectors. Our techniques also yield asymptotic results: Suppose one runs Lanczos on a sequence of Hermitian matrices $A_n \in M_n(\mathbb{C})$ whose spectral distributions converge in Kolmogorov distance with rate $O(n^{-\varepsilon})$ to a density $\mu$ for some $\varepsilon > 0$. Then we show that for large enough $n$, and for $k=O(\sqrt{\log n})$, the Jacobi coefficients output after $k$ iterations concentrate around those for $\mu$. The asymptotic setting is relevant since Lanczos is often used to approximate the spectral density of an infinite-dimensional operator by way of the Jacobi coefficients; our result provides some theoretical justification for this approach. In a different direction, we show that Lanczos fails with high probability to identify outliers of the spectrum when run for at most $c' \log n$ iterations, where again $c'$ depends only on the same global property of the spectrum of $A$. Classical results imply that the bound $c' \log n$ is tight up to a constant factor.