The strong law of large numbers when the mean is undefined

Erickson, K. Bruce

doi:10.1090/S0002-9947-1973-0336806-5

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.

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