The sub-Gaussian norm of a binary random variable
Authors:
V. V. Buldygin and K. K. Moskvichova
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 86 (2013), 33-49
MSC (2010):
Primary 60G50, 65B10, 60G15; Secondary 40A05
DOI:
https://doi.org/10.1090/S0094-9000-2013-00887-4
Published electronically:
August 20, 2013
MathSciNet review:
2986448
Full-text PDF Free Access
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Additional Information
Abstract: The exact values of the sub-Gaussian norms of Bernoulli random variables and binary random variables are found. Exponential bounds for the distributions of sums of centered binary random variables are studied for both cases of independent and dependent random variables. These bounds improve some known results.
References
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- E. I. Ostrovskiĭ, Exponential estimates of the distribution of the maximum of a non-Gaussian random field, Teor. Veroyatnost. i Primenen. 35 (1990), no. 3, 482–493 (Russian); English transl., Theory Probab. Appl. 35 (1990), no. 3, 487–499 (1991). MR 1091204, DOI https://doi.org/10.1137/1135068
- V. V. Petrov, A generalization and sharpening of Bernšteĭn’s inequalities, Vestnik Leningrad. Univ. 22 (1967), no. 19, 63–68 (Russian, with English summary). MR 0219111
- Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press, Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441
- Herman Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statistics 23 (1952), 493–507. MR 57518, DOI https://doi.org/10.1214/aoms/1177729330
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References
- V. V. Buldygin, Skhodimost sluchainykh elementov v topologicheskikh prostranstvakh, “Naukova Dumka”, Kiev, 1980 (Russian). MR 734899 (84m:60011)
- V. V. Buldygin and Ju. V. Kozačenko, Sub-Gaussian random variables, Ukrain. Mat. Zh. 32 (1980), no. 6, 723–730 (Russian). MR 598605 (82d:60023)
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716 (2001g:60089)
- V. V. Buldygin and E. D. Pechuk, Inequalities for the distributions of functionals of sub-Gaussian vectors, Teor. Imovirnost. Matem. Statist. 80 (2009), 23–33; English transl. in Theor. Probability and Math. Statist. 80 (2010), 25–36.
- S. N. Bernstein, On a modification of Chebyshev’s inequality and of the error formula of Laplace, Uchenye Zapiski Nauch.-Issled. Kaf. Ukraine, Sect. Math. (1924), no. 1, 38–48. (Russian)
- S. N. Bernšteĭn, Sobranie sochinenii. Tom IV: Teoriya veroyatnostei. Matematicheskaya statistika. 1911–1946, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0169758 (30 \#2)
- N. N. Vakhaniya, V. V. Kvaratskheliya, and V. I. Tarieladze, Weakly sub-Gaussian random elements in Banach spaces, Ukraïn. Mat. Zh. 57 (2005), no. 9, 1187–1208 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 57 (2005), no. 9, 1387–1412. MR 2216040 (2007c:60006), DOI https://doi.org/10.1007/s11253-006-0003-y
- Yu. V. Kozachenko, Sufficient conditions for the continuity with probability one of sub-Gaussian stochastic processes, Dopovidi Academy of Science of Ukraine, 2 (1968), 113–115. (Ukrainian)
- E. I. Ostrovskiĭ, Exponential estimates of the distribution of the maximum of a non-Gaussian random field, Teor. Veroyatnost. i Primenen. 35 (1990), no. 3, 482–493 (Russian); English transl., Theory Probab. Appl. 35 (1990), no. 3, 487–499 (1991). MR 1091204 (92f:60086), DOI https://doi.org/10.1137/1135068
- V. V. Petrov, A generalization and sharpening of Bernšteĭn’s inequalities, Vestnik Leningrad. Univ. 22 (1967), no. 19, 63–68 (Russian, with English summary). MR 0219111 (36 \#2194)
- Valentin V. Petrov, Limit theorems of probability theory, Oxford Studies in Probability, vol. 4, The Clarendon Press Oxford University Press, New York, 1995. Sequences of independent random variables; Oxford Science Publications. MR 1353441 (96h:60048)
- Herman Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statistics 23 (1952), 493–507. MR 0057518 (15,241c)
- R. Giuliano Antonini and Yu. V. Kozachenko, A note on the asymptotic behavior of sequences of generalized subGaussian random vectors, Random Oper. Stochastic Equations 13 (2005), no. 1, 39–52. MR 2130246 (2006b:60020), DOI https://doi.org/10.1163/1569397053300900
- Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30. MR 0144363 (26 \#1908)
- Masashi Okamoto, Some inequalities relating to the partial sum of binomial probabilities, Ann. Inst. Statist. Math. 10 (1958), 29–35. MR 0099733 (20 \#6171)
- J.-P. Kahane, Propriétés locales des fonctions à séries de Fourier aléatoires, Studia Math. 19 (1960), 1–25 (French). MR 0117506 (22 \#8285)
- Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR 0254888 (40 \#8095)
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Additional Information
V. V. Buldygin
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “KPI”, Peremogy Avenue, 37, Kyiv 03056, Ukraine
Email:
matan@kpi.ua
K. K. Moskvichova
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “KPI”, Peremogy Avenue, 37, Kyiv 03056, Ukraine
Email:
matan@kpi.ua
Keywords:
Sub-Gaussian random variable,
sub-Gaussian norm,
exponential inequalities,
sums of random variables,
Bernstein inequality,
Hoeffding inequality
Received by editor(s):
October 10, 2011
Published electronically:
August 20, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
