Convergence Of Probability Measures Research Articles

On Value Distribution of Certain Beurling Zeta-Functions

In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the system of generalized integers NP generated by P satisfies ∑m⩽x,m∈N1=ax+O(xδ), a>0, 0⩽δ<1, and the function ζP(s) in some strip lying in σ^<σ<1, σ^>δ, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces.

Mean-Field Limit for a Class of Stochastic Ergodic Control Problems

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 23 October 2020Accepted: 18 October 2021Published online: 14 February 2022Keywordsmean-field control, ergodic optimal control, McKean--Vlasov limit, de Finetti's theorem, strong Kac's chaos, convergence of probability measures on path spaceAMS Subject Headings49N80, 60H10, 60F15Publication DataISSN (print): 0363-0129ISSN (online): 1095-7138Publisher: Society for Industrial and Applied MathematicsCODEN: sjcodc

Elipsinių kreivių L funkcijų išvestinės diskretusis universalumas

Straipsnyje įrodyta elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo teorema silpnojo tikimybinių matų konvergavimo prasme analizinių funkcijų erdvėje. Nagrinėjamas analizinės funkcijos aproksimavimas postūmiais L‘E (s + imh), čia kompleksinio kintamojo menamosios dalies postūmiai įgyja reikšmes iš diskrečiosios aibės, pavyzdžiui, aritmetinės progresijos. Fiksuotas skaičius h > 0 pasirenkamas taip, kad exp{2πk/h} būtų iracionalusis skaičius visiems k ∈ Z \{0} . Elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo įrodymas remiasi diskrečiąja ribine teorema tikimybinių matų silpnojo konvergavimo prasme analizinių funkcijų erdvėje.

A Weighted Universality Theorem for L-Functions of Elliptic Curves

In the paper, a short survey on universality results for L-functions of elliptic curves over the field of rational numbers is given and weighted universality theorem is proven. All stated universality theorems are of continuous type. The proof of the universality for L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

On the Generalizations of Universality Theorem for L-Functions of Elliptic Curves

In the paper, the continuous type’s universality theorem for L-functions of elliptic curves is discussed and its generalizations in three directions – for positive integer powers and derivatives of L-functions of elliptic curves as well as the weighted universality theorem of L-functions of elliptic curves – are given. The proofs of the universality are based on limit theorems in the sense of weak convergence of probability measures in functional spaces.

A discrete limit theorem for the periodic Hurwitz zeta-function. II

In the paper, we prove a limit theorem of discrete type on the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta-function.

From convergence in distribution to uniform convergence

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of $n$-by-$n$ matrices as $n$ goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szeg\H{o} type. Our results transfer these convergence theorems into uniform convergence statements.

A discrete limit theorem for the periodic Hurwitz zeta-function

In the paper, we prove a limit theorem of discrete type on the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta-function.

A new method for generating approximation algorithms for financial mathematics applications

This paper describes a new technique that can be used in financial mathematics for a wide range of situations where the calculation of complicated integrals is required. The numerical schemes proposed here are deterministic in nature but their proof relies on known results from probability theory regarding the weak convergence of probability measures. We adapt those results to unbounded payoffs under certain mild assumptions that are satisfied in finance. Because our approximation schemes avoid repeated simulations and provide computational savings, they can potentially be used when calculating simultaneously the price of several derivatives contingent on the same underlying. We show how to apply the new methods to calculate the price of spread options and American call options on a stock paying a known dividend. The method proves useful for calculations related to the log-Weibull model proposed recently for empirical asset pricing.

Binary input reconstruction for linear systems: A performance analysis

Recovering the digital input of a time-discrete linear system from its (noisy) output is a significant challenge in the fields of data transmission, deconvolution, channel equalization, and inverse modeling. A variety of algorithms have been developed for this purpose in the last decades, addressed to different models and performance/complexity requirements. In this paper, we implement a straightforward algorithm to reconstruct the binary input of a one-dimensional linear system with known probabilistic properties. Although suboptimal, this algorithm presents two main advantages: it works online (given the current output measurement, it decodes the current input bit) and has very low complexity. Moreover, we can theoretically analyze its performance: using results on convergence of probability measures, Markov processes, and Iterated Random Functions we evaluate its long-time behavior in terms of mean square error.

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ρ′-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ρ′-mixing stationary random field, due to the strong mixing condition, doesn’t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(−∞, ∞).

On the convergence of Shannon differential entropy, and its connections with density and entropy estimation

This work extends the study of convergence properties of the Shannon differential entropy, and its connections with the convergence of probability measures in the sense of total variation and direct and reverse information divergence. The results relate the topics of distribution (density) estimation, and Shannon information measures estimation, with special focus on the case of differential entropy. On the application side, this work presents an explicit analysis of the density estimation, and differential entropy estimation, for distributions defined on a finite-dimension Euclidean space (Rd,B(Rd)). New consistency results are derived for several histogram-based estimators: the classical product scheme, the Barron's estimator, one of the approaches proposed by Györfi and Van der Meulen, and the data-driven partition scheme of Lugosi and Nobel.

On the value distribution of L-functions of the Selberg class

In this paper, two limit theorems in the sense of the weak convergence of probability measures in the space of meromorphic functions and on the complex plane are proved for L-functions from a subclass of the Selberg class. The explicit form of the limit measures is given.

A generalization of the Curtiss theorem for moment generating functions

The Curtiss theorem deals with the relation between the weak convergence of probability measures on the line and the convergence of theirmoment generating functions in a neighborhood of zero. We present a multidimensional generalization of this result. To this end, we consider arbitrary σ-finite measures whose moment generating functions exist in a domain of multidimensional Euclidean space not necessarily containing zero. We also prove the corresponding converse statement.

JOINT WEIGHTED LIMIT THEOREMS FOR GENERAL DIRICHLET SERIES

In the paper,two joint weighted limit theorems in the sense of weak convergence of probability measures on the complex plane for general Dirichlet series are obtained. The first of them gives only the existence of the limit measure, while in the second theorem,under some additional hypothesis on the weight function, the explicit form of the limit measure is presented. Namely, the limit measure coincides with the distribution of some random element related to considered Dirichlet series.

Value-distribution of twisted L-functions of normalized cusp forms

A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L-functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.

On convergence determining and separating classes of functions

Herein, we generalize and extend some standard results on the separation and convergence of probability measures. We use homeomorphism-based methods and work on incomplete metric spaces, Skorokhod spaces, Lusin spaces or general topological spaces. Our contributions are twofold: we dramatically simplify the proofs of several basic results in weak convergence theory and, concurrently, extend these results to apply more immediately in a number of settings, including on Lusin spaces.

Limit theorems for the Mellin transform of the fourth power of the Riemann zeta-function

We prove limit theorems in the sense of weak convergence of probability measures on the complex plane as well as in the space of analytic functions for the Mellin transform of {ie88-01}

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.

A two-dimensional limit theorem for Mellin transforms of the Riemann zeta-function

In this paper, we obtain a two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for Mellin transforms of the square and the fourth power of the Riemann zeta-function.