Abstract
Limit theorems for the maximum of cumulative sums of stable random variables are discussed. Some new results are obtained for the Cauchy distribution.
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Literature cited
D. A. Darling, “The maximum of sums of stable random variables,” Trans. Am. Math. Soc.,83, No. 1, 164–169 (1956).
C. C. Heyde, “On the maximum of sums of random variables and the supremum functional for stable processes,” J. Appl. Prob.,6, No. 2, 519–429 (1969).
V. B. Nevzorov, “On joint distributions of random variables connected with fluctuations of a stable process,” Teor. Veroyatn. Primen.,18, No. 1, 164–172 (1973).
A. A. Pecherskii and B. A. Rogozin, “On joint distributions of random variables connected with fluctuations of a process with independent increments,” Teor. Veroyatn. Primen.,14, No. 3, 431–444 (1969).
P. Erdös and M. Kac, “On certain limit theorems in the theory of probability,” Bull. Am. Math. Soc.,52, 292–302 (1946).
V. B. Nevzorov, “On some estimates for the distribution of the maximum of cumulative sums,” Teor. Veroyatn. Mat. Statist., No. 5, 88–97 (1971).
T. V. Arak, “On the distribution of cumulative sums of independent random variables,” Teor. Veroyatn. Primen.,19, No. 2, 257–277 (1974).
C. C. Heyde, “A limit theorem for random walks with drift,” J. Appl. Prob.,4, No. 1, 144–150 (1967).
B. A. Rogozin, “The rate of convergence of the distribution of the maximum of sums of independent random variables to a limit law,” Teor. Veroyatn. Primen.,11, No. 3, 497–500 (1966).
V. Paulauskas and S. Steinshunas, “On the rate of convergence of the distribution of the maximum of cumulative sums of independent, identically distributed random vectors to a limit law,” Litov. Mat. Sb.,13, No. 1, 139–147 (1973).
V. B. Nevzorov, “On the nearness of the distribution of the maximum of cumulative sums of independent terms to the distribution of the last sum,” Teor. Veroyatn. Mat. Statist., No. 8, 106–109 (1973).
A. A. Borovkov, “On factorization identities and properties of the distribution of the supremum of cumulative successive sums,” Teor. Veroyatn. Primen.,15, No. 3, 377–418 (1970).
A. K. Aleshkyavichene, “Approximation of the distribution of the maximum of sums of random variables with a mean that is small in absolute value,” Dokl. Akad. Nauk SSSR,233, No. 6, 1017–1020 (1977).
V. B. Nevzorov, “On a class of limit laws for the maximum of cumulative sums,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,72, 92–97 (1977).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 169–174, 1979.
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Nevzorov, V.B. Maximum of cumulative sums for the Cauchy distribution. J Math Sci 20, 2221–2224 (1982). https://doi.org/10.1007/BF01240000
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DOI: https://doi.org/10.1007/BF01240000