Let Sₙ = X₁ + ⋯ + Xₙ where \ Xₙ\ are i.i.d. random variables with EX₁^ ± = ∞. An integral test is given for each of the three possible alternatives lim (Sₙ/n) = + ∞ a.s.; lim (Sₙ/n) = - ∞ a.s.; lim (Sₙ/n) = + ∞ and lim inf (Sₙ/n) = - ∞ a.s. Some applications are noted.
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K. Bruce Erickson (1973) studied this question.
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