The strong law of large numbers when the mean is undefined
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- by K. Bruce Erickson PDF
- Trans. Amer. Math. Soc. 185 (1973), 371-381 Request permission
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.References
- K. G. Binmore and Melvin Katz, A note on the strong law of large numbers, Bull. Amer. Math. Soc. 74 (1968), 941โ943. MR 230354, DOI 10.1090/S0002-9904-1968-12098-1
- C. Derman and H. Robbins, The strong law of large numbers when the first moment does not exist, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 586โ587. MR 70873, DOI 10.1073/pnas.41.8.586
- K. Bruce Erickson, A renewal theorem for distributions on $R^{1}$ without expectation, Bull. Amer. Math. Soc. 77 (1971), 406โ410. MR 279906, DOI 10.1090/S0002-9904-1971-12717-9
- William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- Harry Kesten, The limit points of a normalized random walk, Ann. Math. Statist. 41 (1970), 1173โ1205. MR 266315, DOI 10.1214/aoms/1177696894
- Simon Kochen and Charles Stone, A note on the Borel-Cantelli lemma, Illinois J. Math. 8 (1964), 248โ251. MR 161355
- John A. Williamson, Fluctuations when $E(X_{1})=\infty$, Ann. Math. Statist. 41 (1970), 865โ875. MR 264731, DOI 10.1214/aoms/1177696964
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 185 (1973), 371-381
- MSC: Primary 60G50; Secondary 60F15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0336806-5
- MathSciNet review: 0336806