The central limit theorem II

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Some Limit Theorems for Large Deviations

Contents Introduction 1. Formulation of the problem 2. Survey of results 3. Basic results 4. Contents of the article §1. The phase space of a Markov chain 1. The space Y 2. Metric on Y 3. Group action of G on Y 4. Characteristic measures on Y §2. Strong law of large numbers 1. μ-random walk on Y 2. Ergodicity of the μ-random walk 3. Non-equality of Lyapunov exponents 4. Estimates of z(yg(n)) 5. Proof of Theorem 0.1 6. Rate of contraction to a point §3. Limit theorems for Markov chains 1. The Ionescu-Tulcea and Marinescu theorem 2. Perturbed Markov operators 3. Decomposition of Pη(τ) for small τ 4. Central limit theorem 5. Local limit theorem §4. Proof of the central and local limit theorems §5. Proof of the conditional limit theorem 1. Properties of the operator Kβ 2. Properties of the operator Pβ(τ) 3. Critical case 4. Proof of the conditional limit theorem Bibliography

A universal result in almost sure central limit theory

Uniform Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem

Central and local limit theorems for the coefficients of polynomials of binomial type

On the Rate of Convergence in the Central Limit Theorem in Certain Banach Spaces

Previous article Next article Nonclassical Estimates of the Rate of Convergence in the Central Limit Theorem Taking into Account Large DeviationsS. Ya. ShorginS. Ya. Shorginhttps://doi.org/10.1137/1127034PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Sazonov, A new general estimate of the rate of convergence in the central limit theorem in $R^{k}$, Proc. Nat. Acad. Sci. USA, 71 (1974), 118–121 0277.60010 CrossrefGoogle Scholar[2] V. I. Rotar', Non-classical estimates of the rate of convergence in the multi-dimensional central limit theorem. I, Theory Prob. Appl., 22 (1977), 755–772 0391.60026 LinkGoogle Scholar[3] V. I. Rotar', Non-classical estimates for the rate of convergence in the multidimensional central limit theorem. II, Theory Prob. Appl., 23 (1978), 50–62 0423.60021 LinkGoogle Scholar[4] S. V. Nagaev and , V. I. Rotar', Sharpening of Lyapunov type estimates (the case when the distributions of the summands are close to the normal distribution), Theory Prob. Appl., 18 (1973), 107–119 LinkGoogle Scholar[5] S. Ya. Shorgin, A non-classical estimate of the rate of convergence in the multi-dimensional central limit theorem taking into account large deviations, Theory Prob. Appl., 23 (1978), 667–671 LinkGoogle Scholar[6] S. V. Nagaev and , S. K. Sakoyan, An estimate for the probability of large deviations, Limit theorems and mathematical statistics (Russian), Izdat. “Fan” Uzbek. SSR, Tashkent, 1976, 132–140, 190 55:9237 Google Scholar[7] S. K. Sakoyan, Estimates for distribution functions of sums of random variables taking into account large deviations, 1977, author's abstract of dissertation, Tashkent Google Scholar[8] V. I. Rotar', Masters Thesis, Limit Theorems for Linear and Multilinear Forms, dissertation, LGU, Moscow-Leningrad, 1978 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Limit Theorems on Large DeviationsLimit Theorems of Probability Theory | 1 Jan 2000 Cross Ref Probabilistic Inequalities for Sums of Independent RandomVariables in Terms of Truncated PseudomomentsS. V. NagaevTheory of Probability & Its Applications, Vol. 42, No. 3 | 17 February 2012AbstractPDF (265 KB)On the Exit of a Random Walk From a Curvilinear BoundaryM. U. Gafurov and V. I. Rotar’Theory of Probability & Its Applications, Vol. 28, No. 1 | 17 July 2006AbstractPDF (477 KB) Volume 27, Issue 2| 1983Theory of Probability & Its Applications215-440 History Submitted:26 March 1980Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1127034Article page range:pp. 324-337ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the Accuracy of Normal Approximation of the Probability of Hitting a Ball

Notations and abbreviations 1. Some basic concepts and theorems of probability theory 2. Probability inequalities for sums of independent random variables 3. Weak limit-theorems: convergence to infinitely divisible distributions 4. Weak limit-theorems: the central limit theorem and the weak law of large numbers 5. Rates of convergence in the central limit theorem 6. Strong limit theorems: the strong law of large numbers 7. Strong limit theorems: the law of the iterated logarithm References Author index Subject index

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

In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers (LLN), central limit theorem (CLT), and law of the iterated logarithm (LIL) play a critical role in probability theory. For uncertain variables, basic and important research is also to obtain the relevant limit theorems. However, up to now, there has been no research on these limit theorems for uncertain variables. The main results to emerge from this paper are a strong law of large numbers (SLLN), a weak law of large numbers (WLLN), a CLT, and an LIL for Bernoulli uncertain sequence. For studying these theorems, we first propose an assumption, which can be regarded as a generalization of the duality axiom for uncertain measure in the case that the uncertainty space can be finitely partitioned. Additionally, several new notions such as weakly dependent, Bernoulli uncertain sequence, and continuity from below or continuity from above of uncertain measure are introduced. As far as we know, this is the first study of the LLN, the CLT, and the LIL for uncertain variables. All the theorems proved in this paper can be applied to uncertain variables with symmetric or asymmetric distributions. In particular, the limit of uncertain variables is symmetric in (c) of the third theorem, and the asymptotic distribution of uncertain variables in the fifth theorem is symmetrical.

Chapter 7 - Dynamic quantum Bernoulli random walks

It is shown that (to the extent that the moments involved exist) the existence (≠0) of all (generalized) integral scales is necessary (and sufficient if all moments exist) for integrals over adjacent segments of a stationary process to become asymptotically independent, and sufficient to ensure that existing moments of integrals will become Gaussian. The conditions under which several recent central limit and related theorems for dependent variables have been proven, are shown to be closely related to this requirement. As a consequence of this examination, a slight weakening is suggested of the common condition that the spectrum be non-zero. Several physical problems are described, which may be resolved by the application of such a central limit theorem: longitudinal dispersion in a channel flow (previously treated semi-empirically); the spreading of hot spots, or the expansion of macromolecules; the weak interaction hypothesis (of Kraichnan) for Fourier components. Finally, it is shown that dispersion in homogeneous turbulence is unlikely to be explicable on the basis of a central limit theorem.KeywordsCentral Limit TheoremGaussian ProcessAdjacent SegmentIntegral ScaleHomogeneous TurbulenceThese 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.

In this paper we shall give a brief review of recent results on both the central limit theorems and the non-central limit theorems for non-linear functions of a stationary Gaussian process.