Rates of convergence for partial mass problems

We consider a class of partial mass problems in which a fraction of the mass of a probability measure is allowed to be changed (trimmed) to maximize fit to a given pattern. This includes the problem of optimal partial transportation of mass, where a part of the mass need not be transported, and also trimming procedures which are often used in statistical data analysis to discard outliers in a sample (the data with lowest agreement to a certain pattern). This results in a modified, trimmed version of the original probability which is closer to the pattern. We focus on the case of the empirical measure and analyze to what extent its optimally trimmed version is closer to the true random generator in terms of rates of convergence. We deal with probabilities on ${\mathbb{R}^k}$ and measure agreement through probability metrics. Our choices include transportation cost metrics, associated to optimal partial transportation, and the Kolmogorov distance. We show that partial transportation (as opposed to classical, complete transportation) results in a sharp decrease of costs only in low dimension. In contrast, for the Kolmogorov metric this decrease is seen in any dimension.     

Scenario reduction in stochastic programming

 Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy. Received: July 2000 / Accepted: May 2002 Published online: February 14, 2003 Key words. stochastic programming – quantitative stability – Fortet-Mourier metrics – scenario reduction – transportation problem – electrical load scenario tree Mathematics Subject Classification (1991): 90C15, 90C31     

Stability of invariant measures and continuity of the KdV flow

We prove continuity of the KdV flow on spaces of probability measures with respect to some Wasserstein metrics on H^s, s > 0 and obtain, as a consequence, stability of the invariant measure built by Bourgain in [2]. The main reference is paper [7], joint work with A.S. de Suzzoni.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Preservation of rates of convergence under mappings

For appropriate metrics characterizing various modes of stochastic convergence, it is shown that rates of convergence are preserved by a large class of functions. For example, the extensions of a Lipschitz function on a separable metric space S to the space of all probability measures on S with the Prohorov metric and to the space of all S-valued random variables with the usual metric associated with convergence in probability inherit the Lipschitz property. Consequently, just as with the continuous mapping theorem associated with ordinary convergence, new rate of convergence theorems can sometimes be obtained from old ones by applying appropriate mappings.     

CLTs and Asymptotic Variance of Time-Sampled Markov Chains

For a Markov transition kernel P and a probability distribution μ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel $P_{\mu} = \sum_k \mu(k)P^k.$ In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker’s and Metropolis algorithms in terms of asymptotic variance.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

Non-uniform bounds in local limit theorems in case of fractional moments. I

Edgeworth-type expansions for convolutions of probability densities and powers of the characteristic functions with non-uniform error terms are established for i.i.d. random variables with finite (fractional) moments of order s ≥ 2, where s may be noninteger.     

Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation??s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ?? _??, then the limit is a scale mixture of ?? _??. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

An adelic Hankel summation formula

In this paper, the convergence of the Euler product of the Hecke zeta-function ζ(s,χ) is proved on the line R(s)=1 with s≠1. A certain functional identity between ζ(s,χ) and ζ(2−s,χ) is found. An analogue of Tate's adelic Poisson summation is obtained for the global Hankel transformation, which is constructed in Li (2010) [7].     

Asymptotics of optimal quantizers for some scalar distributions

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.     

Overlaps help: Improved bounds for group testing with interval queries

Given a finite ordered set of items and an unknown distinguished subset P of up to p positive elements, identify the items in P by asking the least number of queries of the type ‘‘does the subset Q intersect P?”, where Q is a subset of consecutive elements of {1,2,…,n}. This problem arises, e.g., in computational biology, in a particular method for determining splice sites in genes. We consider time-efficient algorithms where queries are arranged in a fixed number s of stages: In each stage, queries are performed in parallel. In a recent bioinformatics paper, we proved optimality (subject to lower-order terms) with respect to the number of queries, of some strategies for the special cases p=1 or s=2. Exploiting new ideas, we are now able to provide improved lower bounds for any p?2 and s?3 and improved upper bounds for larger s. Most notably, our new bounds converge as s grows. Our new query scheme uses overlapping query intervals within a stage, which is effective for large enough s. This contrasts with our previous results for s?2 where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small s?3. Anyway, the remaining gaps between the current upper and lower bounds for any fixed s?3 amount to small constant factors in the main term. The paper ends with a discussion of practical implications in the case that the positive elements are well separated.     

Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

Let A be a compact set in Rp of Hausdorff dimension d. For s∈(0,d), the Riesz s-equilibrium measure μs,A is the unique Borel probability measure with support in A that minimizes the double integral over the Riesz s-kernel |x−y|^−s over all such probability measures. In this paper we show that if A is a strictly self-similar d-fractal, then μs,A converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s approaches d from below.     

A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes

This paper provides new necessary and sufficient conditions for a Gaussian random field to have a Gohberg-Krein representation in terms of an n-parameter Wiener process (n > 1). As an application, it demonstrates the nonexistence of a Gohberg-Krein representation of W_s,t ? stW_1,1 in terms of the two-parameter Wiener process W_s,t with (s, t) ? [0, a] × [0, b] for 0 < a < 1, 0 < b < 1.     

On the block thresholding wavelet estimators with censored data

We consider block thresholding wavelet-based density estimators with randomly right-censored data and investigate their asymptotic convergence rates. Unlike for the complete data case, the empirical wavelet coefficients are constructed through the Kaplan-Meier estimators of the distribution functions in the censored data case. On the basis of a result of Stute [W. Stute, The central limit theorem under random censorship, Ann. Statist. 23 (1995) 422-439] that approximates the Kaplan-Meier integrals as averages of i.i.d. random variables with a certain rate in probability, we can show that these wavelet empirical coefficients can be approximated by averages of i.i.d. random variables with a certain error rate in L². Therefore we can show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes , p≥2, q≥1 and nearly optimal convergence rates when 1≤p<2. We also show that these estimators achieve optimal convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of random censoring, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes. The performance of the estimators is tested via a modest simulation study.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

Ratio limit theorems for branching Ornstein-Uhlenbeck processes

We prove ratio limit theorems for critical ano supercritical branching Ornstein-Uhlenbeck processes. A finite first moment of the offspring distribution {p_n} assures convergence in probability for supercritical processes and conditional convergence in probability for critical processes. If even Σp_nnlog⁺log⁺n< ∞, then almost sure convergence obtains in the supercritical case.     

Well-posedness for the Cauchy problem associated to the Hirota-Satsuma equation: Periodic case

We consider a system of Korteweg-de Vries (KdV) equations coupled through nonlinear terms, called the Hirota-Satsuma system. We study the initial value problem (IVP) associated to this system in the periodic case, for given data in Sobolev spaces Hs×Hs+1 with regularity below the one given by the conservation laws. Using the Fourier transform restriction norm method, we prove local well-posedness whenever s>−1/2. Also, with some restriction on the parameters of the system, we use the recent technique introduced by Colliander et al., called I-method and almost conserved quantities, to prove global well-posedness for s>−3/14.     

Increasing the convergence rate of series

This paper deals with the question of obtaining from the sequence {s_n} of partial sums of a convergent series s a new sequence {t_n} which converges to the same limit s as s_n, but more rapidly. When the general term u_n of the series s possesses certain types of expansion involving inverse powers of n, it is shown how t_n is obtained by adding a fixed number of terms to s_n. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make t_n much more rapidly convergent to s—in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.