Abstract
This article presents a number of classical limit theorems for sums of independent random variables and more recent results which are closely related to the classical theorems. It concentrates on three basic subjects: the central limit theorem, the laws of large numbers and the law of the iterated logarithm for sequences of independent real-valued random variables. The author was restricted to an article of small size. Therefore many chapters of the classical theory of summation of independent random variables were omitted, particularly limit theorems with non-normal limit distributions, multidimensional limit theorems and local limit theorems. This article may be regarded as an introduction for the reader who wishes to become acquainted with the classical limit theorems for sums of independent random variables without spending much time. More detailed presentations may be found, for example, in the author’s book (1987) and in its predecessors Csörgö and Révész (1981), Gnedenko and Kolmogorov (1949), Hall (1982), Ibragimov and Linnik (1965), Loève (1960), Petrov (1972, 1995), Révész (1967) and Stout (1974).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Baum, L.E., and Katz. M. (1965): Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, No. 1, 108–123. Zbl. 142. 14802
Berkes, I. (1972): A remark to the law of the iterated logarithm. Studia Sci. Math. Hungar. 7, No. 1–2, 189–197. Zbl. 265. 60025
Berry, A.C. (1941): The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49, No. 1, 122–136. Zbl. 025. 34603
Bikelis, A. (1966): Estimates of the remainder term in the central limit theorem. Lit. Mat. Sb. 6, No. 3 (in Russian), 323–346. Zbl. 149. 14002
Buldygin, V.V. (1978): The strong law of large numbers and convergence of Gaussian sequences to zero. Teor. Veroyatn. Mat. Stat., Kiev, No. 19, 33–41. Engl. transl.: Theory Probab. Math. Stat. 19, 35–43 (1980; Zbl. 485.60029). Zbl 407. 60021
Csörgö, M., and Révész, P. (1981): Strong Approximations in Probability and Statistics. Akadémiai Kiadd, Budapest. Zbl. 539. 60029
Egorov, V.A. (1969): On the law of the iterated logarithm. Teor. Veroyatn. Primen. 14, No. 4. 722–729. Engl. transi.: Theor. Probab. Appl. 14, No. 4, 693–699. Zbl. 211. 48903
Egorov, V.A. (1972): On Kolmogorov’s theorem on the law of the iterated logarithm. Vestn. Leningr. Univ. Ser I 13, 140–142 ( Russian ). Zbl. 244. 6022
Egorov, V.A. (1973): On the rate of convergence to normal law which is equivalent to the existence of a second moment. Teor. Veroyatn. Primen. 18, No. 1, 180–185. Engl. transi.: Theory Probab. Appl. 18, No. 1, 175–180. Zbl. 307. 60025
Esseen, C.G. (1942): On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astron. Fys. A28, No. 2, 1–19. Zbl. 027. 33902
Esseen, C.G. (1945): Fourier analysis of distribution functions. Acta Math. 77, 1–125. Zbl. 060. 28705
Etemadi, N. (1981): An elementary proof of the strong law of large numbers. Z. Wahrschlichkeitstheorie Verw. Geb. 55, No. 1, 119–122. Zbl. 448. 60024
Feller, W. (1968): On the Berry—Esseen theorem. Z. Wahrscheinlichkeitstheorie Verw. Geb. 10, No. 3, 261–268. Zbl. 167. 17304
Gnedenko, B.V., and Kolmogorov, A.N. (1949): Limit Distributions for Sums of Independent Random Variables. Gostekhizdat, Moscow-Leningrad. Engl. transl.: Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, Mass. (1954). Zbl. 056. 36001
Hall, P. (1980): Characterizing the rate of convergence in the central limit theorem. Ann. Probab. 8, No. 6, 1037–1048. Zbl. 456. 60018
Hall, P. (1982): Rates of Convergence in the Central Limit Theorem. Pitman, Boston, London and Melbourne. Zbl. 497. 60001
Heyde, C.C. (1967): On the influence of moments on the rate of convergence to the normal distribution. Z. Wahrscheinlichkeitstheorie Verw. Geb. 8, No. 1, 12–18. Zb. 149. 14001
Heyde, C.C. (1969): Some properties of metrics in a study of convergence to normality. Z. Wahrscheinlichkeitstheorie Verw. Geb. 11, No. 3, 181–192. Zbl. 169. 20902
Heyde, C.C. (1973): On the uniform metric in the context of convergence to normal- ity. Z. Wahrscheinlichkeitstheorie Verw. Geb. 25, No. 2, 83–95. Zbl. 699. 62095
Heyde, C.C., and Nakata, T. (1984): On the asymptotic equivalence of L 7, metrics for the convergence to normality. Z. Wahrscheinlichkeitstheorie Verw. Geb. 68, No. 1, 97–106. Zbl. 546. 60035
Ibragimov, I.A. (1966): On the approximation of distribution functions of sums of independent variables. Teor. Veroyatn. Primen. 11, No. 4, 632–655. Engl. transl.: Theor. Probab. Appl. 11, No. 4, 559–580. Zbl. 161. 15207
Ibragimov, I.A., and Linnik, Yu.V. (1965): Independent and Stationarily Connected Sequences of Variables. Nauka, Moscow. Zbl. 154.42201. Engl. transi.: Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen (1971). Zbl. 219. 60027
Katz, M. (1963): Note on the Berry—Esseen theorem. Ann. Math. Stat. 34, No. 3, 1107–1108. Zbl. 122. 36704
Klass, M.J., and Tomkins, R.J. (1984): On the limiting behavior of normed sums of independent variables. Z. Wahrscheinlichkeitstheorie Verw. Geb. 68, No. 1, 107–120, Zbl. 552. 60026
Komlôs, J., Major, P., and Tusnâdy, G. (1975): An approximation of partial sums of independent RV’s, and the sample DF, I, II. Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, No. 1–2, 111–131; 34, No. 1, 33–58. Zbl. 308.60029, Zbl. 315. 60031
Kruglov, V.M. (1976): Global limit theorems. Zap. Nauch. Semin. Leningr. Otd. Mat. Inst. Steklova 61, 84–191. Engl. transi.: J. Sov. Math. 16, 1396–1409. Zbl. 358. 60036
Loève, M. (1960): Probability Theory. 2nd ed., Van Nostrand, Pinceton, NJ. Zbl. 095. 12201
Major, P. (1976a): The approximation of partial sums of independent RV’s. Z. Wahrscheinlichkeitstheorie Verw. Geb. 35, No. 3, 213–220. Zbl. 338. 60031
Major, P. (1976b): Approximation of partial sums of i.i.d.r.v’s when the summands have only two moments. Z. Wahrscheinlichkeitstheorie Verw. Geb. 35, No. 3, 221–229. Zbl. 338. 60032
Major, P. (1979): An improvement of Strassen’s invariance principle. Ann. Probab. 7, No. 1, 55–61. Zbl. 392. 60034
Marcinkiewicz, J.,and Zygmund, A. (1937): Remarque sur la loi du logarithme itéré. Fundam. Math. 29, 215–222. Zbl. 018.03204
Martikainen, A.I. (1979a): Three theorems on the limit superior of sums of independent random variables. Vestn. Leningr. Univ., Mat. Meth. Astron., No. 1, 45–51. Engl. transl.: Vestn. Leningr. Univ. Math. 12, 29–36. Zbl. 411. 60048
Martikainen, A.I. (1979b): An exponential criterion for the law of the iterated logarithm. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 85, 158–168. Engl. transl.: J. Soy. Math. 20, 2214–2221 (1982). Zbl. 417. 60042
Martikainen, A.I. (1979c): On necessary and sufficient conditions for the strong law of large numbers. Teor. Veroyatn. Primen. 24, No. 4, 814–821. Engl. transl.: Theory Probab. Appl. 24, No. 4, 813–820. Zbl. 432. 60036
Martikainen, A.I. (1980): A converse to the law of the iterated logarithm for a random walk. Teor. Veroyatn. Primen. 25, No. 2, 364–366. Engl. transi.: Theory Probab. Appl. 25, No. 2, 361–362. Zbl. 432. 60037
Martikainen, A.I., and Petrov, V.V. (1977): On necessary and sufficient conditions for the law of the iterated logarithm. Teor. Veroyatn. Primen. 22, No. 1, 18–26; No. 2, 442. Engl. transi.: Theor. Probab. Appl. 22, No. 1, 16–23; No. 2, 430. Zbl. 377. 60036
Nagaev, S.V. (1965): Some limit theorems for large deviations. Teor. Veroyatn. Primen. 10, No. 2, 231–254. Engl. transi.: Theor. Probab. Appl. 10, No. 2, 214–235. Zbl. 144. 18704
Nagaev, S.V. (1972): On necessary and sufficient conditions for the strong law of large numbers. Teor. Veroyatn. Primen. 17, No. 4, 609–618. Engl. transl.: Theor. Probab. Appl. 17, No. 4, 573–581. Zbl. 276. 60035
Osipov, L.V. (1966): Refinement of Lindeberg’s theorem. Teor. Veroyatn. Primen. 11, No. 2, 339–342. Engl. transl.: Theor. Probab. Appl. 11, No. 2, 299–302. Zbl. 147. 37001
Osipov, L.V. (1967): Asymptotic expansions in the central limit theorem. Vestn. Leningr. Univ., Ser I 19, 45–62 (in Russian). Zbl. 189. 18003
Osipov, L.V. (1968): On the closeness with which the distribution of the sum of independent random variables approximates the normal distribution. Dokl. Akad. Nauk SSSR 178, No. 5, 1013–1016. Engl. transl.: Sov. Math., Dokl. 9, No. 1, 233236. Zbl. 185. 46803
Osipov, L.V. (1971): Asymptotic expansions for the distributions of sums of independent random variables. Teor. Veroyatn. Primen. 16, No. 2, 328–338. Engl. transl.: Theor. Probab. App. 16, No. 2, 333–343. Zbl. 248. 60015
Osipov, L.V., and Petrov, V.V. (1967): On an estimate of the remainder term in the central limit theorem. Teor. Veroyatn. Primen. 12, No. 2, 322–329. Engl. transl.: Theor. Probab. Appl. 12, No. 2, 281–286. Zbl. 185. 46801
Petrov, V.V. (1962): On some polynomials encountered in probability theory. Vestn. Leningr. Univ., Ser I 19, 150–153 (in Russian). Zbl. 128. 38003
Petrov, V.V. (1965): An estimate of the deviation of the distribution of a sum of independent random variables from the normal law. Dokl. Akad. Nauk 160, No. 5, 1013–1015. Engl. transl.: Soy. Math., Dokl. 6, No. 1, 242–244. Zbl. 135. 19203
Petrov, V.V. (1966): On a relation between an estimate of the remainder in the central limit theorem and the law of the iterated logarithm. Teor. Veroyatn. Primen. 11, No. 3, 514–518. Engl. transl.: Teor. Probab. Appl. 11, No. 3, 454–458. Zbl. 203. 19602
Petrov, V.V. (1969): On the strong law of large numbers. Teor. Veroyatn. Primen. 14, No. 2, 193–202. Engl. transl.: Theor. Probab. Appl. 14, No. 2, 183–192. Zbl. 196. 20903
Petrov, V.V. (1972): Sums of Independent Random Variables. Nauka, Moscow. Engl. transl.: Springer-Verlag, Berlin-Heidelberg — New York (1975). Zbl. 267. 60055 (Zbl. 322.60042)
Petrov, V.V. (1987): Limit Theorems for Sums of Independent Random Variables. Nauka, Moscow (in Russian). Zbl. 621. 60022
Petrov, V.V. (1995): Limit Theorems of Probability Theory. Oxford University Press, Oxford. Zbl. 826–60357
Prokhorov, Yu.V. (1950): On the strong law of large numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 14, No. 6, 523–536 (in Russian). Zbl. 040. 07301
Prokhorov, Yu.V. (1959): Some remarks on the strong law of large numbers. Teor. Veroyatn. Primen. 4, No. 2, 215–220. Engl. transl.: Theor. Probab. Appl. 4, No. 2, 204–208. Zbl. 089. 13903
Pruitt, W.E. (1981): General one-sided laws of the iterated logarithm. Ann. Probab. 9, No. 1, 1–48. Zbl. 462. 60030
Révész, P. (1967): The Laws of Large Numbers. Akadémiai Kiad6, Budapest and NewYork, Academic Press, 1968. Zbl. 203. 50403
Rosalsky, A. (1980): On the converse to the iterated logarithm law. Sankhya A42, No. 1–2, 103–108. Zbl. 486. 60031
Rozovskii, L.V. (1975): Asymptotic expansions in the central limit theorem. Teor. Veroyatn. Primen. 20, No. 4, 810–820. Engl. transi.: Theory Probabl. Appl. 20, Mno. 4, 794–804. Zbl. 347. 60020
Rozovskii, L.V. (1978a): On a lower bound for the remainder in the central limit theorem. Mat Zametki 24, No. 3, 403–410. Engl. transi.: Math. Notes 24, No. 3–4, 715–719. Zbl. 396. 60026
Rozovskii, L.V. (1978b): On the exactness of an estimate of the remainder term in the central limit theorem. Teor. Veroyatn. Primen. 24, No. 4, 744–761. Engl. transi.: Theory Probabl. Appl. 23, No. 4, 712–730. Zbl. 388. 60026
Sakhanenko, A.I. (1984): Rate of convergence in an invariance principle for non-identically distributed variables with exponential moments. Tru. Inst. Mat. 3, 4–49. Engl. transl.: Advances in Pribability Theory, Transi. for Math. Engl. 2–73 (1986). Zbl. 541. 60024
Sakhanenko, A.I. (1985): Estimates in an invariance principle, Tru. Inst. Mat. 5, 27–44 (in Russian). Zbl. 585. 60044
Stout, W.F. (1974): Almost Sure Convergence, Academic Press, New York. Zbl. 321. 60022
Strassen, V. (1964): An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie Verw. Geb. 3, No. 3, 211–226. Zbl. 132. 12903
Strassen, V. (1966): A converse to the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie Verw. Geb. 4, 265–268. Zbl. 141. 16501
Tomkins, R.J. (1980): Limit theorems without moment hypotheses for sums of in- dependent random variables. Ann. Probab. 8, No. 2, 314–324. Zbl. 432. 60034
Tomkins, R.J. (1991): Refinements of Kolmogorov’s law of the iterated logarithm. Stat. Probab. Lett. 14, No. 4, 321–325 (1992), Zbl. 756. 60033
Weiss, M. (1959): On the law of the iterated logarithm. J. Math. Mech. 8, No. 1, 121–132. Zbl. 091. 14206
Zolotarev, V.M. (1986): Modern Theory of Summation of Independent Random Variables. Nauka, Moscow (in Russian). Engl. transi.: VSP, Utrecht (1997). Zbl. 649. 60016
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Petrov, V.V. (2000). Classical-Type Limit Theorems for Sums of Independent Random Variables. In: Prokhorov, Y.V., Statulevičius, V. (eds) Limit Theorems of Probability Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04172-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-662-04172-7_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08170-5
Online ISBN: 978-3-662-04172-7
eBook Packages: Springer Book Archive