The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces

We extend to random fields case, the results of Woyczynski, who proved Brunk′s type strong law of large numbers (SLLNs) for 𝔹‐valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above‐mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

?Strong laws of large numbers which are useful in the theory and applications of stochastic processes are obtained for sequences of independent random elements in separable normed linear spaces. The hypotheses for these results lie between those for the identically distributed case and the independent non-identieally distributed case. These results and other strong and weak laws of large numbers for separable normed linear spaces can be extended to separable Freshet spa?es. Finally, the results are applied to separable Wiener processes on [0, 1] and on [0, oo).

Probability Inequalities for Sums of Independent Random Variables

: Strong laws of large numbers for a sequence x sub n of random functions in D(0,1) are derived using new pointwise conditions on the first absolute moments, which improve on known results. In particular, convex tightness is not implied by the hypotheses of the theorems. It is shown that convex tightness is is preserved when random functions are centered, and this result is applied to improve some known strong laws for weighted sums in D(0,1). A weak law of large numbers is proved using a new pointwise condition on the first moments and some weak laws for weighted sums are improved upon by weakening the hypotheses. A study is made of relationships among several conditions on X sub n which appear as hypotheses in laws of large numbers. (Author)

On Cantrell–Rosalsky’s strong laws of large numbers

Previous article Next article On the Dependence of the Convergence Rate in the Strong Law of Large Numbers for Stationary Processes on the Rate of Decay of the Correlation FunctionV. F. GaposhkinV. F. Gaposhkinhttps://doi.org/10.1137/1126078PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] André Blanc-Lapierre and , Albert Tortrat, Sur la loi forte des grands nombres pour les fonctions aléatoires stationnaires du second ordre, C. R Acad. Sci. Paris Sér. A-B, 267 (1968), A740–A743 39:4924 0177.46103 Google Scholar[2] I. N. Verbitskaya, On conditions for the applicability of the strong law of large numbers to wide-sense stationary processes, Theory Prob. Appl., 11 (1966), 632–636 LinkGoogle Scholar[3] V. V. Petrov, The strong law of large numbers for a stationary sequence, Dokl. Akad. Nauk SSSR, 213 (1973), 42–44, (In Russian.) 49:4083 0303.60032 Google Scholar[4] R. J. Serfling, Moment inequalities for the maximum cumulative sum, Ann. Math. Statist., 41 (1970), 1227–1234 42:3835 0272.60013 CrossrefGoogle Scholar[5] V. F. Gaposhkin, Convergence of series that are connected with stationary sequences, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 1366–1392, 1438, (In Russian.) 53:6694 Google Scholar[6] V. F. Gaposhkin, Criteria for the strong law of large numbers for some classes of second-order stationary processes and homogeneous random fields, Theory Prob. Appl., 22 (1977), 286–310 0377.60033 LinkGoogle Scholar[7] V. F. Gaposhkin, Exact estimates of the rate of convergence in the strong law of large numbers for classes of stationary (in the wide sense) sequences and processes, Uspehi Mat. Nauk, 31 (1976), 233–234, (In Russian.) 55:9240 Google Scholar[8] V. F. Gaposhkin, Estimates of means for almost all realizations of stationary processes, Sibirsk. Mat. Zh., 20 (1979), 978–989, 1165, (In Russian.) 82d:60064 0447.60028 Google Scholar[9] G. Aleksich, Problems in the Convergence of Orthogonal Series, IL, Moscow, 1963, (In Russian.) Google Scholar[10] A. Zygmund, Trigonometrical Series, 2 vols., Cambridge University Press, Cambridge, 1959 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Exponent of Convergence of a Sequence of Ergodic Averages26 August 2022 | Mathematical Notes, Vol. 112, No. 1-2 Cross Ref Weighted laws of large numbers and convergence of weighted ergodic averages on vector valued $$L_p$$-spaces5 July 2021 | Advances in Operator Theory, Vol. 6, No. 3 Cross Ref Constructive approach to limit theorems for recurrent diffusive random walks on a stripAsymptotic Analysis, Vol. 122, No. 3-4 Cross Ref Almost everywhere convergence of ergodic series6 October 2015 | Ergodic Theory and Dynamical Systems, Vol. 37, No. 2 Cross Ref Pointwise equidistribution with an error rate and with respect to unbounded functions4 April 2016 | Mathematische Annalen, Vol. 367, No. 1-2 Cross Ref An effective Ratner equidistribution result for SL(2,R)⋉R2Duke Mathematical Journal, Vol. 164, No. 5 Cross Ref On Estimates of the Convergence Rate in the Strong Law of Large Numbers for Stationary Sequences (in Some Classes $S_w$ and $R_\Phi$)V. F. Gaposhkin10 November 2011 | Theory of Probability & Its Applications, Vol. 55, No. 4AbstractPDF (194 KB)POINTWISE ERGODIC THEOREMS WITH RATE WITH APPLICATIONS TO LIMIT THEOREMS FOR STATIONARY PROCESSES21 November 2011 | Stochastics and Dynamics, Vol. 11, No. 01 Cross Ref Pointwise ergodic theorems with rate and application to the CLT for Markov chainsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 45, No. 3 Cross Ref Extensions of the Menchoff-Rademacher theorem with applications to ergodic theoryIsrael Journal of Mathematics, Vol. 148, No. 1 Cross Ref Fractional Poisson equations and ergodic theorems for fractional coboundariesIsrael Journal of Mathematics, Vol. 123, No. 1 Cross Ref Volume 26, Issue 4| 1982Theory of Probability & Its Applications History Submitted:18 December 1979Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126078Article page range:pp. 706-720ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

In this paper, we shall present weak and strong laws of large numbers (WLLN and SLLN) for weighted sums of independent (not necessarily identically distributed) fuzzy set-valued random variables in the sense of the extended Hausdorff metric d H ∞ , based on the results of set-valued random variable obtained by Taylor and Inoue [34], [35]. This work is a continuation of [21].

Purpose The aim of this work is to prove a strong law of large numbers for a sequence of independent compactly uniformly integrable random sets with values in the family of convex closed subsets of a separable Banach space E, again without requiring any geometric conditions on E. Design/methodology/approach Our approach in this work is based on several theories; probability and its application to strong law of large numbers, properties of random sets, convex analysis and functional analysis. Findings This article establishes two strong laws of large numbers for independent, compactly uniformly integrable random sets. Originality/value This paper presents original results concerning the Strong Law of Large Numbers (SLLN) for random sets, specifically focusing on compactly uniformly integrable random sets in separable Banach spaces.

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

In this paper, we shall prove the strong law of large numbers (SLLN) for set-valued random variables in the sense of dH, and the basic space being Rademacher type p(1lesples2) Banach space. This kind of SLLN is the extension of classical SLLN's for Xi-valued random variables and it also implies previous SLLN's results for set-valued random variables

An extension of Feller’s strong law of large numbers

Strong law of large numbers and Chover's law of the iterated logarithm under sub-linear expectations

Moment conditions in strong laws of large numbers for multiple sums and random measures

Next article On Necessary and Sufficient Conditions for the Strong Law of Large NumbersS. V. NagaevS. V. Nagaevhttps://doi.org/10.1137/1117072PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. V. Prokhorov, Strong stability of sums and infinitely divisible distributions, Theory Prob. Applications, 3 (1958), 141–153 10.1137/1103013 LinkGoogle Scholar[2] Yu. V. Prohorov, On the strong law of large numbers, Izvestiya Akad. Nauk SSSR. Ser. Mat., 14 (1950), 523–536, (In Russian.) MR0038592 (12,425c) Google Scholar[3] Yu. V. Prokhorov, Some remarks on the strong law of large numbers, Theor. Probability Appl., 4 (1959), 204–208 10.1137/1104018 MR0121858 (22:12588) 0089.13903 LinkGoogle Scholar[4] Pál Révész, The laws of large numbers, Probability and Mathematical Statistics, Vol. 4, Academic Press, New York, 1968, 176–, London MR0245079 (39:6391) 0203.50403 Google Scholar[5] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963xvi+685 MR0203748 (34:3596) 0108.14202 Google Scholar Next article FiguresRelatedReferencesCited byDetails A Generalization of the Kolmogorov Theorem on the Law of the Iterated LogarithmL. V. Rozovskii17 February 2012 | Theory of Probability & Its Applications, Vol. 42, No. 1AbstractPDF (121 KB)Limit Theorems for Independent Random Variables Cross Ref Almost sure behaviour ofF-valued random fieldsProbability Theory and Related Fields, Vol. 93, No. 3 Cross Ref The Strong Law of Large Numbers Cross Ref A Note on the Strong Law of Large Numbers for Partial Sums of Independent Random Vectors**Most of this paper has been written while the author was working at the Fachbereich Mathematik WE 1, Freie Universität Berlin. Cross Ref The Work of A. N. Komogorov on Strong Limit TheoremsN. H. Bingham17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 1AbstractPDF (1280 KB)Sur la loi des grands nombres de Nagaev en dimension infinie Cross Ref A Law of Large Numbers for Random Vectors having Large Norms Cross Ref Limit Theorems for Independent Random Variables Cross Ref Book Reviews28 June 2008 | Australian Journal of Statistics, Vol. 23, No. 1 Cross Ref On Necessary and Sufficient Conditions for the Strong Law of Large NumbersA. I. Martikainen17 July 2006 | Theory of Probability & Its Applications, Vol. 24, No. 4AbstractPDF (726 KB)A Remark on the Strong Law of Large NumbersN. A. Volodin and S. V. Nagaev17 July 2006 | Theory of Probability & Its Applications, Vol. 22, No. 4AbstractPDF (333 KB)Limit Theorems for Independent Random Variables Cross Ref On the Strong Law of Large NumbersS. V. Nagaev and N. A. Volodin17 July 2006 | Theory of Probability & Its Applications, Vol. 20, No. 3AbstractPDF (402 KB)The Behavior of Sums of Independent Random VariablesV. M. Kruglov28 July 2006 | Theory of Probability & Its Applications, Vol. 19, No. 2AbstractPDF (506 KB) Volume 17, Issue 4| 1973Theory of Probability & Its Applications History Submitted:12 April 1971Published online:28 July 2006 InformationCopyright © 1973 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1117072Article page range:pp. 573-581ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Previous article Next article On Necessary and Sufficient Conditions for the Law of the Iterated LogarithmA. I. Martikainen and V. V. PetrovA. I. Martikainen and V. V. Petrovhttps://doi.org/10.1137/1122002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Petrov, Certain theorems of the type of the law of the iterated logarithm, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 41 (1974), 129–132, (In Russian.) MR0365673 Google Scholar[2] Leonard E. Baum, , M. Katz and , H. H. Stratton, Strong laws for ruled sums, Ann. Math. Statist., 42 (1971), 625–629 MR0290427 0217.21101 CrossrefGoogle Scholar[3] V. V. Petrov, On a generalization of a Levy inequality, Theory Prob. Applications, 20 (1975), 141–145 10.1137/1120012 0349.60001 LinkGoogle Scholar[4] V. V. Petrov, Sums of Independent Random Variables, Nauka, Moscow, 1972, (In Russian.) Google Scholar[5] Yu. V. Prohorov, On the strong law of large numbers, Izvestiya Akad. Nauk SSSR. Ser. Mat., 14 (1950), 523–536, (In Russian.) MR0038592 Google Scholar[6] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963xvi+685 MR0203748 0108.14202 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Spitzer Series and Regularly Varying FunctionsPseudo-Regularly Varying Functions and Generalized Renewal Processes | 13 October 2018 Cross Ref A Generalization of the Kolmogorov Theorem on the Law of the Iterated LogarithmTheory of Probability & Its Applications, Vol. 42, No. 1 | 17 February 2012AbstractPDF (121 KB)On the Strong Law of Large Numbers for Random Quadratic FormsTheory of Probability & Its Applications, Vol. 40, No. 1 | 28 July 2006AbstractPDF (1480 KB)Rates of convergence for tail seriesJournal of Statistical Planning and Inference, Vol. 43, No. 1-2 | 1 Jan 1995 Cross Ref On Strong Limit Theorems for Sums of Independent Random VariablesTheory of Probability & Its Applications, Vol. 37, No. 1 | 28 July 2006AbstractPDF (463 KB)On Convergence of Sums of Operator-Normed Independent Random VectorsTheory of Probability & Its Applications, Vol. 36, No. 2 | 17 July 2006AbstractPDF (568 KB)The Strong Law of Large Numbers for Sums of Independent Random Vectors with Operator Normalizations and Null Convergence of Gaussian SequencesTheory of Probability & Its Applications, Vol. 32, No. 2 | 17 July 2006AbstractPDF (1225 KB)The Contraction Principle and the Strong Law of Large Numbers for Weighted SumsTheory of Probability & Its Applications, Vol. 31, No. 3 | 28 July 2006AbstractPDF (1191 KB)On the One-Sided Law of the Iterated LogarithmTheory of Probability & Its Applications, Vol. 30, No. 4 | 28 July 2006AbstractPDF (910 KB)Summary of Reports Presented at Sessions of the Seminar in Probability Theory and Mathematical Statistics at the Shevchenko State University, KievTheory of Probability & Its Applications, Vol. 30, No. 3 | 17 July 2006AbstractPDF (1098 KB)Criteria for Strong Convergence of Normalized Sums of Independent Random Variables and Their ApplicationsTheory of Probability & Its Applications, Vol. 29, No. 3 | 17 July 2006AbstractPDF (1300 KB)One-Sided Versions of Strong Limit TheoremsTheory of Probability & Its Applications, Vol. 28, No. 1 | 17 July 2006AbstractPDF (1252 KB)On the limiting behavior of normed sums of independent random variablesZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 68, No. 1 | 1 Jan 1984 Cross Ref On the Convergence Rate in the Strong Law of Large NumbersTheory of Probability & Its Applications, Vol. 26, No. 1 | 17 July 2006AbstractPDF (486 KB)On the law of the iterated logarithm in the infinite variance caseJournal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, Vol. 30, No. 1 | 9 April 2009 Cross Ref On Necessary and Sufficient Conditions for the Strong Law of Large NumbersTheory of Probability & Its Applications, Vol. 24, No. 4 | 17 July 2006AbstractPDF (726 KB) Volume 22, Issue 1| 1977Theory of Probability & Its Applications1-202 History Submitted:16 December 1975Published online:17 July 2006 InformationCopyright © 1977 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1122002Article page range:pp. 16-23ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics