Prohorov Distance Research Articles

Survey of Distances between the Most Popular Distributions

We present a number of upper and lower bounds for the total variation distances between the most popular probability distributions. In particular, some estimates of the total variation distances in the cases of multivariate Gaussian distributions, Poisson distributions, binomial distributions, between a binomial and a Poisson distribution, and also in the case of negative binomial distributions are given. Next, the estimations of Lévy–Prohorov distance in terms of Wasserstein metrics are discussed, and Fréchet, Wasserstein and Hellinger distances for multivariate Gaussian distributions are evaluated. Some novel context-sensitive distances are introduced and a number of bounds mimicking the classical results from the information theory are proved.

Distance between metric measure spaces and distance matrix distributions

We study the Prohorov distance between the distance matrix distributions of two metric measure spaces. We prove that it is not smaller than 1-box distance between two metric measure spaces and also prove that it is not larger than 0-box distance between two metric measure spaces.

Distributional convergence in planar dynamics and singular perturbations*1

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.

Explicit bounds on Levy–Prohorov distance for a class of multidimensional distribution functions

Let F( x 1,…, x k ) and G( x 1,…, x k )= F X 1 ( x 1)… F X k ( x k ), where F X i (x i), 1⩽i⩽k , are the one-dimensional marginal distributions of F, be two distribution functions on R k . Here, we obtain explicit bounds for the Levy–Prohorov distance between F and G using some general results due to Yurinskii (1975, Theory Probab. Appl. 20, 1–10).

A note on nearby variables with nearby conditional laws

We apply a version of the Strassen–Dudley theorem for Markov kernels to obtain a strong approximation theorem for sequences of random variables with values in Polish spaces. The conditions are expressed in terms of the Prohorov distance of regular conditional distributions. By avoiding discrete approximation of the random variables we can improve the upper bounds for the approximation.

Metric marginal problems for set-valued or non-measurable variables

In a separable metric space, if two Borel probability measures (laws) are nearby in a suitable metric, then there exist random variables with those laws which are nearby in probability. Specifically, by a well-known theorem of Strassen, the Prohorov distance between two laws is the infimum of Ky Fan distances of random variables with those laws. The present paper considers possible extensions of Strassen's theorem to two random elements one of which may be (compact) set-valued and/or non-measurable. There are positive results in finite-dimensional spaces, but with factors depending on the dimension. Examples show that such factors cannot entirely be avoided, so that the extension of Strassen's theorem to the present situation fails in infinite dimensions.

A note on the prohorov distance between a counting process and a poisson process

In 1991, Nikunen and Valkeila obtained an upper bound for the Prohorov distance between an arbitrary counting process and a Poisson process in terms of their compensators, where the compensator of the Poisson process was strictly increasing and continuous. The aim of this note is to generalize this result to the case where the compensator of the Poisson process is only continuous.

A prohorov bound for a poisson process and an arbitrary counting process with some applications

An upper bound for the Prohorov distance between a Poission process and an arbitrary couting process in terms of their compensators is examined. We discuss the sharpness of this bound and apply it to the known weak convergense results of Brown [1] and Kabanov, Liptser and Shiryayev [2]. We also use our bound to investigate empirical and renewal counting processes.

About the prohorov distance between the uniform distribution over the unit cube in R dand its empirical measure

We compute the almost sure order of convergence of the Prokhorov distance between the uniform distribution P over [0, 1] d and the empirical measure associated with n independent observations with (common) distribution P. We show that this order of convergence is n -1/d up to a power of log(n). This result extends to the case where the observations are weakly dependent.

A sharpening of the remainder term in the higher-dimensional central limit theorem for multilinear rank statistics

A new approach to the asymptotic normality of the multivariate linear rank statistics is provided along with the Berry-Esséen and the Prohorov distance estimates for the remainder term in the convergence to normality.

Strong approximation of very weak Bernoulli processes

Very weak Bernoulli processes with values in a separable metric space are introduced. An estimate for the Prohorov distance in the central limit theorem is obtained. This estimate is used to establish a strong (almost sure) approximation of the partial sums of a very weak Bernoulli process by a Brownian motion where the error term is of the order O(t1/2−γ). The proofs are based on a new version of the Berkes-Philipp approximation theorem.

Robustness of Estimators on Stationary Observations

Hampel's general qualitative definition of robustness of sequences of estimators on memoryless observation processes is generalized to stationary ergodic processes by substituting the generalized Ornstein (or $\bar{\phi}$) distance for the marginal Prohorov distance as the measure of "closeness" of observations. More general sequences of estimators are also allowed. The approach yields results analogous to those of Hampel for the more general case considered, often provides strict generalizations of Hampel's results, and in some cases yields simpler proofs.

On the quality of poisson approximations

Poisson processes (possibly nonhomogeneous) are constructed in the function spaces D q ≡D([0, 1] q , R) and Dq q x ⋯ x D q in order to approximate superpositions of uniformly sparse point processes and partial sums of infinitesimal integer-valued nonnegative random variables. Bounds for the Prohorov distance are computed, where the Prohorov distance is defined on the space of all probability measures on D q, with the Skorohod metric being used on D q. These bounds yield functional central limit theorems (invariance principles) and rates of convergence for functional central limit theorems involving convergence to the Poisson process. In this regard, this paper is an extension of Section 6 of Dudley [4].