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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The strong law of large numbers when the mean is undefined


Author: K. Bruce Erickson
Journal: Trans. Amer. Math. Soc. 185 (1973), 371-381
MSC: Primary 60G50; Secondary 60F15
MathSciNet review: 0336806
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Abstract: Let $ {S_n} = {X_1} + \cdots + {X_n}$ where $ \{ {X_n}\} $ are i.i.d. random variables with $ EX_1^ \pm = \infty $. An integral test is given for each of the three possible alternatives $ \lim ({S_n}/n) = + \infty $ a.s.; $ \lim ({S_n}/n) = - \infty $ a.s.; $ \lim \sup ({S_n}/n) = + \infty $ and $ \lim \inf ({S_n}/n) = - \infty $ a.s. Some applications are noted.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0336806-5
Keywords: Independent identically distributed random variables, mean undefined, strong law of large numbers, renewal function, truncated mean function
Article copyright: © Copyright 1973 American Mathematical Society