A limit theorem for sums of independent random elements in a Banach space
Author:
I. K. Matsak
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 100 (2020), 141-152
MSC (2010):
Primary 60B12
DOI:
https://doi.org/10.1090/tpms/1102
Published electronically:
August 5, 2020
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Additional Information
Abstract: Conditions for the convergence of the maximal norm of sums of identically distributed random elements in Banach spaces are studied. Some applications are discussed for $\omega ^2$ type statistics.
References
- L. Bachelier, Théorie de la spéculation, Ann. Sci. École Norm. Sup. (3) 17 (1900), 21–86 (French). MR 1508978
- P. Erdös and M. Kac, On certain limit theorems of the theory of probability, Bull. Amer. Math. Soc. 52 (1946), 292–302. MR 15705, DOI https://doi.org/10.1090/S0002-9904-1946-08560-2
- A. V. Skorohod and N. P. Slobodenjuk, Predel′nye teoremy dlya sluchaĭ nykh bluzhdaniĭ, Izdat. “Naukova Dumka”, Kiev, 1970 (Russian). MR 0282419
- V. I. Paulauskas, The distribution of the maximum of the successive sums of independent identically distributed random variables, Litovsk. Mat. Sb. 13 (1973), no. 2, 133–138, 263 (Russian, with Lithuanian and English summaries). MR 0339318
- V. Paulauskas and S. Steišūnas, The rate of convergence of the distribution of the maximum of the successive sums of independent nonidentically distributed random vectors to the limit law, Litovsk. Mat. Sb. 13 (1973), no. 2, 139–147, 263 (Russian, with Lithuanian and English summaries). MR 0339319
- Ī. K. Matsak, Some limit theorems for the maximum of sums of independent random processes, Ukraïn. Mat. Zh. 60 (2008), no. 12, 1664–1674 (Ukrainian, with Russian summary); English transl., Ukrainian Math. J. 60 (2008), no. 12, 1955–1967. MR 2523114, DOI https://doi.org/10.1007/s11253-009-0183-3
- Ī. K. Matsak, A. M. Plīchko, and A. S. Sheludenko, Limit theorems for the maximum of sums of independent random processes, Ukraïn. Mat. Zh. 70 (2018), no. 4, 506–518 (Ukrainian, with English and Russian summaries); English transl., Ukrainian Math. J. 70 (2018), no. 4, 581–596. MR 3805391, DOI https://doi.org/10.1007/s11253-018-1518-8
- N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR 1435288
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015
- X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, École d’Été de Probabilités de Saint-Flour, IV-1974, Springer, Berlin, 1975, pp. 1–96. Lecture Notes in Math., Vol. 480 (French). MR 0413238
- G. Pólya and G. Szegö, Problems and Theorems from Analysis, vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, 1964.
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- J. Lamperty, Probability, Benjamin, New York, 1966.
- Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1965 (French). Suivi d’une note de M. Loève; Deuxième édition revue et augmentée. MR 0190953
- Z. Ciesielski, Hölder conditions for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403–413. MR 132591, DOI https://doi.org/10.1090/S0002-9947-1961-0132591-2
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
- I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, vol. 1, “Nauka”, Moscow, 1971; English transl. Springer-Verlag, Berlin, Heidelberg, 1979.
- L. N. Bol′shev and N. V. Smirnov, Tablitsy matematicheskoĭ statistiki, 3rd ed., “Nauka”, Moscow, 1983 (Russian). With an afterword by Yu. V. Prokhorov and D. M. Chibisov; With a commentary and bibliography by D. S. Shmerling. MR 735434
- A. S. Sheludenko, Limit theorem for some statistics of the Kolmogorov–Smirnov type, Bull. Taras Shevchenko Nat. Univ. Kyiv, ser. Math. Mech. 38 (2017), no. 4, 54–58. (Ukrainian)
References
- P. Bachelier, Théorie de la spéculation, Ann. Sci. de l’É. N. S., $3^{e}$ série, 17 (1900), 21–86. MR 1508978
- E. Erdös and M. Kac, On certain limit theorems in the theory of probability, Bull. Amer. Math. Soc. 52 (1946), 292–302. MR 15705
- A. V. Skorokhod and N. P. Slobodenyuk, Limit Theorems for Random Walks, “Naukova dumka”, Kyiv, 1970. (Russian) MR 0282419
- V. Paulauskas, On the distribution of the maximum of consecutive sums independent identically distributed random vectors, Lietuvos matem. rinkinys 13 (1973), 133–138. MR 0339318
- V. Paulauskas and S. Steishunas, On the rate of convergence of the maximum distribution of consecutive sums of independent random vectors to limit law, Lietuvos matem. rinkinys 13 (1973), 139–147. MR 0339319
- I. K. Matsak, On some limit theorems for the maximum of sums of independent random processes, Ukr. Matem. Zh. 60 (2008), no. 12, 1664–1674; English transl. in Ukrain. Math. J. 60 (2008), no. 12, 1955–1967. MR 2523114
- I. K. Matsak, A. M. Plichko, and A. S. Sheludenko, Limit theorems for the maximum of sums of independent random processes, Ukr. Matem. Zh. 70 (2018), no. 4, 506–518; English transl. in Ukrain. Math. J. 70 (2018), no. 4, 581–596. MR 3805391
- N. N. Vachania, B. I. Tarieladze, and S. A. Chobanyan, Probability Distributions on Banach Spaces, “Nauka”, Moscow, 1985; English transl., D. Reidel Publishing Company, Dordrecht–Boston–Lancaster–Tokyo, 1987. MR 1435288
- M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991. MR 1102015
- X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Lect. Not. Math. 480 (1975), 1–96. MR 0413238
- G. Pólya and G. Szegö, Problems and Theorems from Analysis, vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, 1964.
- P. Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, London, Sydney, Toronto, 1968. MR 0233396
- J. Lamperty, Probability, Benjamin, New York, 1966.
- P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1937. MR 0190953
- Z. Ciesielski, Hölder condition for realizations of Gaussian processes, Trans. Amer. Math. Soc. 99 (1961), 403–413. MR 132591
- J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, 1977. MR 0415253
- I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, vol. 1, “Nauka”, Moscow, 1971; English transl. Springer-Verlag, Berlin, Heidelberg, 1979.
- L. N. Bol’shev and N. V. Smirnov, Tables of Mathematical Statistics, “Nauka”, Moscow, 1983. (Russian) MR 735434
- A. S. Sheludenko, Limit theorem for some statistics of the Kolmogorov–Smirnov type, Bull. Taras Shevchenko Nat. Univ. Kyiv, ser. Math. Mech. 38 (2017), no. 4, 54–58. (Ukrainian)
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Additional Information
I. K. Matsak
Affiliation:
Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 2, Building 6, Kyiv, 03127 Ukraine
Email:
ivanmatsak@univ.kiev.ua
Keywords:
Central limit theorem,
Banach spaces,
maximum of norms of sums
Received by editor(s):
January 1, 2019
Published electronically:
August 5, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
