Probabilities Of Large Deviations For Sums Research Articles

Superlarge Deviation Probabilities for Sums of Independent Random Variables with Exponential Decreasing Distribution

Previous article Next article Superlarge Deviation Probabilities for Sums of Independent Random Variables with Exponential Decreasing DistributionL. V. RozovskyL. V. Rozovskyhttps://doi.org/10.1137/S0040585X9798289XPDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractIn this paper large deviation probabilities of sums of independent identically distributed random variables are studied, whose distribution function has an exponential decreasing tail.[1] L. V. Rozovsky, Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cramér condition, Theory Probab. Appl., 48 (2005), pp. 108–130. TPRBAU 0040-585X LinkGoogle Scholar[2] A. A. Borovkov and and B. A. Rogozin, On the multidimensional central limit theorem, Theory Probab. Appl., 10 (1965), pp. 55–62. TPRBAU 0040-585X LinkGoogle Scholar[3] L. V. Rozovsky, Large-deviation probabilities for some classes of distributions satisfying the Cramér condition, Zap. Nauch. Semin. POMI, 298 (2003), pp. 161–185 (in Russian). ZNSPEV 0136-1244 Google Scholar[4] L. V. Rozovsky, On a lower bound of large-deviation probabilities for the sample mean under the Cramér condition, Zap. Nauch. Semin. POMI, 278 (2001), pp. 208–224 (in Russian); J. Math. Sci. New York, 118 (2003), pp. 5624–5634 (in English). Google Scholar[5] A. A. Borovkov and and A. A. Mogulskii, On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II, Theory Probab. Appl., 51 (2007), pp. 567–594. TPRBAU 0040-585X LinkGoogle ScholarKeywordsindependent random variableslarge deviationsregular varying function Previous article Next article FiguresRelatedReferencesCited byDetails Superlarge Deviation Probabilities for Sums of Independent Random Variables with Exponentially Decreasing Tails. II25 March 2015 | Theory of Probability & Its Applications, Vol. 59, No. 1AbstractPDF (171 KB) Volume 52, Issue 1| 2008Theory of Probability & Its Applications History Submitted:23 November 2005Published online:10 March 2008 InformationCopyright © 2008 Society for Industrial and Applied MathematicsKeywordsindependent random variableslarge deviationsregular varying functionPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X9798289XArticle page range:pp. 167-171ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Probabilities of Large Deviations of Type II Errors for Tests of Kolmogorov and Omega-Square Types

For the Kolmogorov and omega-square tests, strong asymptotics for large deviations of type II error probabilities are obtained in the case of “least favorable alternatives.” Using these asymptotics, type II error probabilities for any sequence of alternatives can readily be estimated. The proofs are based on an exact asymptotic of large deviation probabilities for Gaussian measures in a Hilbert space and on a theorem on large deviation probabilities for sums of independent random vectors in a Banach space. Bibliography: 22 titles.

Estimate of Distributions of Sums of Independent Random Variables

We investigate necessary and sufficient conditions under which one estimate of exponential type is valid for large deviation probabilities of sums of independent identically distributed random variables. Bibliography: 3 titles.

Large-Deviation Probabilities in Banach Spaces

Theorems concerning exact asymptotics of large-deviation probabilities of sums of independent random elements in a Banach space are proved. We consider probabilities of the following events: sums of independed elements belong to balls such that their centers deviate from the origin as the sample size increases. The results are a version of Bentkus' and Rachkauskas' theorems proved for the exteriors of balls centered at the origin. Bibliography: 11 titles.

Lower Bounds on Large Deviation Probabilities for Sums of Independent Random Variables

Lower Bounds on Large Deviation Probabilities for Sums of Independent Random Variables

An asymptotic formula for the Bayes risk in discriminating between two Markov chains

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

An asymptotic formula for the Bayes risk in discriminating between two Markov chains

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say ℙ{M n ⩾ x}, of sums M n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.

Large deviation probabilities for sums of heavy-tailed dependent random vectors*

Necessary and sufficient conditions are given for multidimensional p – stable limit theorems (i.e. theorems on convergence of normalized partial sums of a stationary sequence of random vectors to a non-degenerate strictly p-stable limiting law μ with -regularly varying normalizing sequence ). It is proved that similarly as in the one-dimensional case the conditions for consist of two parts: one responsible for (very weak) mixing properties and another, describing asymptotics of probabilities of large deviations (with a minor additional condition for ). The paper focuses on effective methods of proving such large deviation results

Large deviation probabilities for sums of lattice random vectors with heavy-tailed distribution

Large deviation probabilities for sums of lattice random vectors with heavy-tailed distribution

Estimates for Distributions of Sums Stopped at Markov Time

In the present paper asymptotically right estimates for moments and large deviation probabilities for sums of a random number of random variables are derived. A relatively simple method of obtaining the required estimates based on the properties of stopping times and close to known proofs of the Wald identity is suggested. The demand for such estimates arises in a variety of problems of probability theory. In our opinion they are of independent interest and may be useful in a diversity of applications (see, for instance, [3], [4]). It should be pointed out that deriving inequalities for distributions of maximum of partial sums A. N. Kolmogorov [1] used essentially the ideas of stopping times. In their paper A. N. Kolmogorov and Yu. V. Prokhorov [2] proposed a proof of the Wald identity, permitting an easily removed of the special propositions of Wald, who considered sums of independent and identically distributed summands stopped at the time of the first output exit from the interval. The same ideas form the basis of the considerations below.

Convergence of the series of large-deviation probabilities for sums of independent equally distributed random variables

The series\(\sum\nolimits_{n \geqslant 1} {\tau _n P(|S_n | \geqslant \varepsilon n^a )}\) is studied, where Sn are the sums of independent equally distributed random variables, τn is a sequence of nonnegative numbers, α>0, and ɛ>0 is an arbitrary positive number. For a broad class of sequences τn, the necessary and sufficient conditions are established for the convergence of this series for any ɛ>0.

Limiting values of large deviation probabilities of quadratic statistics

Application of exact Bahadur efficiencies in testing theory or exact inaccuracy rates in estimation theory needs evaluation of large deviation probabilities. Because of the complexity of the expressions, frequently a local limit of the nonlocal measure is considered. Local limits of large deviation probabilities of general quadratic statistics are obtained by relating them to large deviation probabilities of sums of k-dimensional random vectors. The results are applied, e.g., to generalized Cramér-von Mises statistics, including the Anderson-Darling statistic, Neyman's smooth tests, and likelihood ratio tests.

Large deviation probabilities of sums of independent random vectors for certain classes of sets

Large deviation probabilities of sums of independent random vectors for certain classes of sets

Large Deviation Probabilities for Sums of Functions of Mixing Sequences

Previous article Next article Large Deviation Probabilities for Sums of Functions of Mixing SequencesV. T. Dubrovin and D. A. MoskvinV. T. Dubrovin and D. A. Moskvinhttps://doi.org/10.1137/1124098PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. T. Dubrovin and , D. A. Moskvin, Central limit theorem for sums of functions of missing sequences, Theory Prob. Applications, 24 (1979), 560–571 10.1137/1124065 0435.60025 LinkGoogle Scholar[2] V. T. Dubrovin, Large deviations in a central limit theorem for sums of functions of weakly dependent random variablesProbabilistic methods and cybernetics, No. 9 (Russian), Izdat. Kazan. Univ., Kazan, 1971, 34–44, (In Russian.) MR0317376 Google Scholar[3] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 0219.60027 Google Scholar[4] V. V. Petrov, Generalization of Cramér's limit theorem, Uspehi Matem. Nauk (N.S.), 9 (1954), 195–202 MR0065058 Google Scholar[5] V. M. Zolotarev, Several new probabilistic inequalities connected with the Lévy metric, Dokl. Akad. Nauk SSSR, 190 (1970), 1019–1021 MR0256440 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 24, Issue 4| 1980Theory of Probability & Its Applications History Submitted:27 January 1977Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1124098Article page range:pp. 838-845ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On Large Deviation Probabilities for Sums of Independent Random Vectors

On Large Deviation Probabilities for Sums of Independent Random Vectors

Rough Limit Theorems on Large Deviations for Markov Stochastic Processes, II

Previous article Next article Rough Limit Theorems on Large Deviations for Markov Stochastic Processes, IIA. D. Ventsel’A. D. Ventsel’https://doi.org/10.1137/1121062PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] H. Cramér, Sur un nouveau théorème limité de la théorie des probabilités, Act. Sci. et Ind., 736 (1038), 64.0529.01 Google Scholar[2] A. A. Borovkov, Boundary-value problems for random walks and large deviations in function spaces, Theory Prob. Applications, 12 (1967), 575–595 10.1137/1112074 0178.20004 LinkGoogle Scholar[3] A. D. Ventsel' and , M. I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys, 25 (1970), 1–55 10.1070/rm1970v025n01ABEH001254 0297.34053 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Moderate deviations-based importance sampling for stochastic recursive equations17 November 2017 | Advances in Applied Probability, Vol. 49, No. 4 Cross Ref The influence of demographic stochasticity on evolutionary dynamics and stabilityTheoretical Population Biology, Vol. 88 Cross Ref The Canonical Equation of Adaptive Dynamics: A Mathematical ViewSelection, Vol. 2, No. 1-2 Cross Ref Limit Theorems on Large Deviations Cross Ref Some Large Deviations Results in Markov Fluid Models27 July 2009 | Probability in the Engineering and Informational Sciences, Vol. 6, No. 4 Cross Ref Colliding stacks: A large deviations analysis5 July 2007 | Random Structures & Algorithms, Vol. 2, No. 4 Cross Ref Phase transition in an interacting Bose system. An application of the theory of Ventsel' and FreidlinJournal of Statistical Physics, Vol. 61, No. 3-4 Cross Ref A quick simulation method for excessive backlogs in networks of queuesIEEE Transactions on Automatic Control, Vol. 34, No. 1 Cross Ref Quick Simulation of Excessive Backlogs in Networks of Queues Cross Ref Large Deviations Estimates for Systems with Small Noise Effects, and Applications to Stochastic Systems TheoryPaul Dupuis and Harold J. Kushner1 August 2006 | SIAM Journal on Control and Optimization, Vol. 24, No. 5AbstractPDF (2550 KB)A new technique for analyzing large traffic systems1 July 2016 | Advances in Applied Probability, Vol. 18, No. 2 Cross Ref Large deviations from classical paths. Hamiltonian flows as classical limits of quantum flowsCommunications in Mathematical Physics, Vol. 101, No. 2 Cross Ref Estimates of quantum deviations from classical mechanics using large deviation results15 September 2006 Cross Ref Large deviations and rare events in the study of stochastic algorithmsIEEE Transactions on Automatic Control, Vol. 28, No. 9 Cross Ref Rough Limit Theorems on Large Deviations for Markov Stochastic Processes. IVA. D. Venttsel’17 July 2006 | Theory of Probability & Its Applications, Vol. 27, No. 2AbstractPDF (1661 KB)Rough Limit Theorems on Large Deviations for Markov Stochastic Processes. IIIA. D. Ventsel’17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 4AbstractPDF (1563 KB)On Large Deviation Probabilities for Sums of Independent Random VectorsL. V. Osipov17 July 2006 | Theory of Probability & Its Applications, Vol. 23, No. 3AbstractPDF (1498 KB)Remarks on Large Deviations for the $\omega ^2 $ StatisticsA. A. Mogul’skii17 July 2006 | Theory of Probability & Its Applications, Vol. 22, No. 1AbstractPDF (436 KB) Volume 21, Issue 3| 1977Theory of Probability & Its Applications History Submitted:08 October 1974Published online:28 July 2006 InformationCopyright © 1977 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1121062Article page range:pp. 499-512ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On large deviation probabilities for sums of random variables which are attracted to the normal law

For the sequence {xn of independent identically distributed random variable satisrying P{x1>x} x L(x) and P{-x1>x} x L1(x),where p≥2 and L,L1 are functions of slow variation such that L1(x)/L(x)−p as x→∞,it is shown that P{Sn>>ta}∽np{X1>>tn}for certain sequences tn;here

On Large Deviation Probabilities for Sums of Random Variables Which are Attracted to the Normal Law

On Large Deviation Probabilities for Sums of Random Variables Which are Attracted to the Normal Law

On Inequalities for Large Deviations for Certain Statistics

Previous article Next article On Inequalities for Large Deviations for Certain StatisticsV. V. YurinskiiV. V. Yurinskiihttps://doi.org/10.1137/1116039PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. M. Dudley, Weak convergences of probabilities on nonseperable metric spaces and empirical measures on Euclidean spaces, Illinois J. Math., 10 (1966), 109–126 MR0185641 CrossrefGoogle Scholar[2] J. Kiefer, On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm, Pacific J. Math., 11 (1961), 649–660 MR0131885 0119.34904 CrossrefGoogle Scholar[3] V. V. Yurinskii, On an infinite-dimensional version of S. N. Bernshtein's inequalities, Theory Prob. Applications, 15 (1970), 108–109 10.1137/1115008 0216.46601 LinkGoogle Scholar[4] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), 13–30 MR0144363 0127.10602 CrossrefGoogle Scholar[5] A. V. Prokhorov, Masters Thesis, Inequalities for sums of independent multi-dimensional random variables, Cand. Dist., Moscow, 1968, (In Russian.) Google Scholar[6] I. N. Sanov, On the probability of large deviations of random magnitudes, Mat. Sb. N. S., 42 (84) (1957), 11–44, (In Russian.) MR0088087 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails A Nonasymptotic Bound for Deviation of a Multidimensional Empirical Distribution FunctionE. I. Gaivoronskii and E I. Ostrovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 3AbstractPDF (269 KB)An inequality for differences of distribution functionsArchiv der Mathematik, Vol. 54, No. 2 Cross Ref On some inequalities in the theory of uniform distribution, IProceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 64, No. 3 Cross Ref On Large Deviation Probabilities for Sums of Independent Random VectorsL. V. Osipov17 July 2006 | Theory of Probability & Its Applications, Vol. 23, No. 3AbstractPDF (1498 KB)Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (February–October 1974)17 July 2006 | Theory of Probability & Its Applications, Vol. 20, No. 2AbstractPDF (1740 KB) Volume 16, Issue 2| 1971Theory of Probability & Its Applications History Submitted:15 January 1971Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1116039Article page range:pp. 385-387ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics