On the Supremum of Sn/n

Let X1,X2,⋯ be independent and identically distributed. We give a simple proof based on stopping times of the known result that sup ( | X1 + ⋯ + Xn|/n) has a finite expected value if and only if E | X | log⁡ | X| is finite. Whenever E |X| log ⁡|X| = ∞, a simple nonanticipating stopping rule τ, not depending on X, yields E(|X1+ ⋯ + Xτ | /τ) = ∞.