A theorem about the oscillation of sums of independent random variables

Authors:
D. L. Hanson and F. T. Wright

Journal:
Proc. Amer. Math. Soc. **37** (1973), 226-233

MSC:
Primary 60G50; Secondary 60J15

DOI:
https://doi.org/10.1090/S0002-9939-1973-0315779-0

MathSciNet review:
0315779

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be i.i.d. random variables and let . The relationship between the *t*th moment of and the convergence of the series is investigated in this paper. The convergence of the series above when but is related to the oscillation of the sequence and to the oscillation of the sequence about zero.

**[1]**D. L. Hanson and Melvin Katz,*On the oscillation of sums of random variables*, Proc. Amer. Math. Soc.**17**(1966), 864–865. MR**0195122**, https://doi.org/10.1090/S0002-9939-1966-0195122-7**[2]**C. C. Heyde,*Two probability theorems and their application to some first passage problems*, J. Austral. Math. Soc.**4**(1964), 214–222. MR**0183004****[3]**Einar Hille,*Analytic function theory. Vol. 1*, Introduction to Higher Mathematics, Ginn and Company, Boston, 1959. MR**0107692****[4]**M. Rosenblatt,*On the oscillation of sums of random variables*, Trans. Amer. Math. Soc.**72**(1952), 165–178. MR**0045326**, https://doi.org/10.1090/S0002-9947-1952-0045326-3**[5]**Walter L. Smith,*A theorem on functions of characteristic functions and its application to some renewal theoretic random walk problems*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif, 1967, pp. 265–309. MR**0216542****[6]**Frank Spitzer,*A combinatorial lemma and its application to probability theory*, Trans. Amer. Math. Soc.**82**(1956), 323–339. MR**0079851**, https://doi.org/10.1090/S0002-9947-1956-0079851-X

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0315779-0

Keywords:
Oscillation,
oscillation about zero

Article copyright:
© Copyright 1973
American Mathematical Society