2
Many of the fundamental results in probability theory are formulated as limit theorems. Bernoulli’s law of large numbers was formulated as a limit theorem; so was the De Moivre-Laplace theorem, which can fairly be called the origin of a genuine theory of probability and, in particular, which led the way to numerous investigations that clarified the conditions for the validity of the central limit theorem. Poisson’s theorem on the approximation of the binomial distribution by the “Poisson” distribution in the case of rare events was formulated as a limit theorem. After the example of these propositions, and of results on the rapidity of convergence in the De Moivre-Laplace and Poisson theorems, it became clear that in probability it is necessary to deal with various kinds of convergence of distributions, and to establish the rapidity of convergence connected with the introduction of various “natural” measures of the distance between distributions. In the present chapter we shall discuss some general features of the convergence of probability distributions and of the distance between them. In this section we take up questions in the general theory of weak convergence of probability measures in metric spaces. The De Moivre-Laplace theorem, the progenitor of the central limit theorem, finds a natural place in this theory. From §3, it is clear that the method of characteristic functions is one of the most powerful means for proving limit theorems on the weak convergence of probability distributions in R n. In §6, we consider questions of metrizability of weak convergence. Then, in §8, we turn our attention to a different kind of convergence of distributions (stronger than weak convergence), namely convergence in variation. Proofs of the simplest results on the rapidity of convergence in the central limit theorem and Poisson’s theorem will be given in §§10 and 11.KeywordsProbability MeasureCharacteristic FunctionCentral Limit TheoremWeak ConvergenceIndependent Random VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
24A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem
A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem
40On the Chebyshev-Cramér Asymptotic Expansions
On the Chebyshev-Cramér Asymptotic Expansions
20Continuation of Convergence in the Central Limit Theorem
Previous article Next article Continuation of Convergence in the Central Limit TheoremH.-J. Rossberg and G. SiegelH.-J. Rossberg and G. Siegelhttps://doi.org/10.1137/1120095PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975x+346, Berlin MR0388499 0322.60043 CrossrefGoogle Scholar[2] H.-J. Rossberg, On a problem of Kolmogorov concerning the normal distribution, Theory Prob. Applications, 19 (1974), 795–798 10.1137/1119085 0323.60026 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Continuability of local weak limits for record valuesStatistics, Vol. 44, No. 3 Cross Ref Limit Theorems Involving Restricted ConvergenceH.-J. Rossberg28 July 2006 | Theory of Probability & Its Applications, Vol. 39, No. 2AbstractPDF (2035 KB)Vague Convergence for Convolutions and for Infinitely Divisible FunctionsG. Siegel1 August 2006 | Theory of Probability & Its Applications, Vol. 31, No. 1AbstractPDF (783 KB)One-Sided Characterization of the Normal Distribution in the Set of Infinitely Divisible DistributionsH.-J. Rossberg and G. Siegel17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 2AbstractPDF (665 KB)Limit Theorems on Large Deviations in a Narrow ZoneL. V. Rozovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 26, No. 4AbstractPDF (854 KB)Analytic Distribution Functions Cross Ref Continuation of Distribution Functions Cross Ref Limit Theorems for Identically Distributed Summands Asuming the Convergence of the Distribution Functions on a Half AxisH.-J. Rossberg17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 4AbstractPDF (1398 KB)Uniqueness of Symmetric Distribution Functions Defined on Bounded Sets and Its ApplicationsG. Siegel17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 4AbstractPDF (341 KB)On Moderate Deviation Probabilities for Sums of Independent Random VariablesN. N. Amosova17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 4AbstractPDF (453 KB)On Determining an Infinitely Divisible Distribution Function by Its Values on a Half-LineI. A. Ibragimov17 July 2006 | Theory of Probability & Its Applications, Vol. 22, No. 2AbstractPDF (627 KB) Volume 20, Issue 4| 1976Theory of Probability & Its Applications History Submitted:03 July 1974Published online:17 July 2006 InformationCopyright © 1976 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1120095Article page range:pp. 866-868ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics
Read more1Optimisation of Spare Parts Quality Inspection Cost Based on Simulated Annealing and Genetic Algorithm
This paper provides an in-depth study on the impact of spare parts quality inspection on production costs in the electronics industry. By establishing a model based on binomial distribution, central limit theorem and right-hand side hypothesis testing, the costs, profits and losses of enterprises under different testing strategies are analysed. Firstly, for the inspection cost problem of accessory products, the minimum number of inspections is determined by assuming the number of sample defective products using binomial distribution and approximating the probability by De Moivre-Laplace theorem. Second, the simulated annealing algorithm is combined to derive 16 decision alternatives based on the analysis of control variables, and the best decision with the lowest total cost is finally determined. Finally, for the case of increasing number of spare parts and the appearance of semi-finished products, a mathematical model with minimising cost as the objective function is developed and solved using a genetic algorithm to evaluate the total cost under different inspection strategies. The study shows that the model proposed in this paper can reasonably solve the cost optimisation problem in electronic product quality inspection with high efficiency and practicality.
Read more
Limit Theorems for The Sums of Random Variables in A Special Form
In this paper, we consider some functionals of the sums of independent identically distributed random variables. The functionals of the sums are important in probabilistic models and stochastic branching systems. In connection with the application in various probabilistic models and stochastic branching systems, we are interested in the fulfillment of the law of large numbers and the Central limit theorem for these sums. The main hypotheses of the paper are the presence of second order moments of the variables and the fulfillment of the Lindeberg condition is considered. The research object and subject of this paper consists of specially generated random variables using the sums of non-bound random variables. In total, 6 different sums in a special form were studied in the paper and this sum was not previously studied by other scientists. The purpose of the paper is to examine whether these sums in a special form satisfy the terms of the law of large numbers and the Central limit theorem. The main result of the paper is to show that the law of large numbers and the terms of the classical limit theorem are fulfilled in some cases. The results obtained in the paper are of theoretical importance, The Central limit theorem analogues proved here are applications of Lindeberg theorem. The results can be applied to the determination of the fluctuation of immigration branching systems as well as the asymptotic state of autoregression processes. At the same time, from the main results obtained in the paper it can be used in practical lessons conducted on the theory of probability. The results of the paper will be an important guide for young researchers. Important theorems proved in the paper can be used in probability theory, stochastic branching systems and other practical problems.
Read more1µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces
This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence {μn} of probability measures is weakly convergent to a probability measure μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then {μn} is weakly convergent to μ if and only if limn→∞∫Ωg(s)dμn=∫Ωg(s)dμ for every bounded continuous function g: Ω → X. Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.
Read moreThe Linear Topology Associated with Weak Convergence of Probability Measures
This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by statisticians and our presentation gives an alternative view to most standard probability textbooks. In particular, this functional approach clarifies the underlying topological structure of weak convergence. We hope this short note is helpful for those who are interested in weak convergence as well as instructors of measure theoretic probability.
Read more1ESTIMATION OF THE NUMBER OF SUMMANDS OF THE CENTRAL LIMIT THEOREM
The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.
Read more2Limit 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}
Read more61On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables
On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables
39Contributions to the theory of Markov chains. II
Introduction. This is a sequel to my paper [1]. The present developments are largely independent of the previous results except in so far as given in the Appendix. Theorem 1 shows a kind of solidarity among the states of a recurrent class; it generalizes a classical result due to Kolmogorov and permits a classification of recurrent states and classes. In ?2 some relations involving the mean recurrence and first passage times are given. In ??3-5 sequences of random variables associated in a natural way with a Markov chain are studied. Theorem 2 is a generalized ergodic theorem which applies to any recurrent class, positive or null. It turns out that in a null class there is a set of numbers which plays the role of stationary absolute probabilities. In the case of a recurrent random walk with independent, stationary steps these numbers are all equal to one and the result is particularly simple. Theorem 3 shows that the kind of solidarity exhibited in Theorem 1 persists in such a sequence; it leads to the clarification of certain conditions stated by Doblin(2) in connection with his central limit theorem. Using a fundamental idea due to Doblin, the weak and strong laws of large numbers, the central limit theorem, the law of the iterated logarithm, and the limit theorems for the maxima of the associated sequence are proved very simply. Owing to the great simplicity of the method it is the conditions of validity of these limit theorems that should deserve attention. Among other things, we shall show by an example that a certain set of conditions, attributed to Kolmogorov, is in reality not sufficient for the validity of the central limit theorem. Furthermore, conditions of validity for the strong limit theorems and the limit theorems for the maxima are obtained by a rather natural strengthening of corresponding conditions for the weak limit theorems. A word about the connection of these conditions with martingale theory closes the paper. 1. The sequence of random variables {Xn}, n=0, 1, 2, * , forms a denumerable Markov chain with stationary transition probabilities. The states will be denoted by the non-negative integers(3) 0, 1, 2, * * . The n-step transition probability from the state i to the state j will be denoted by P(n) (P(l) = Pij). Thus we have
Read more65Central and local limit theorems for the coefficients of polynomials of binomial type
Central and local limit theorems for the coefficients of polynomials of binomial type
6On weak convergence of probability measures, channel capacity and code error probabilities
The relationship between weak convergence of channel probability measures, channel capacity, and error probability of block codes is examined for memoryless channels with general input and output alphabets. It is shown that channel capacity is a lower semi-continuous function and that every block code with maximal probability of error /spl delta/ for a nominal channel for any /spl epsiv/>0 can be modified such that the modification has a probability of error less than /spl delta/+/spl epsiv/ for all channels in a sufficiently small neighborhood of the nominal channel.
Read more5A limit theorem for the Hurwitz zeta function with an algebraic irrational parameter
A limit theorem in the sense of weak convergence of probability measures in the space of analytic functions for the Hurwitz zeta function with an algebraic irrational parameter is obtained. Bibliography 6 titles.
Read more2On Convergence and Compactness in Variation with a Shift of Discrete Probability Laws
We consider a class of discrete distribution functions whose characteristic functions are separated from zero, i.e., their absolute values are larger than a certain positive constant on the entire real line. This class is rather wide: it contains discrete infinitely divisible distribution functions, functions of lattice distributions whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently, we demonstrated in our study that characteristic functions corresponding to elements of this class admit the Lévy–Khinchin type representation with a nonmonotonic spectral function. Thus, our class is included in the set of quasi-infinitely divisible distribution functions. Using these representations, we also obtained limit theorems and theorems of compactness with convergence in variation for the sequences from this class. This paper is devoted to analogous results concerning convergence and compactness, but with somewhat weakened convergence in variation. Changing the type of convergence notably expands the applicability of the results.
Read more
Copyright 2026 Cactus Communications. All rights reserved.
