Statistics Papers

Document Type

Journal Article

Date of this Version

1970

Publication Source

The Annals of Mathematical Statistics

Volume

41

Issue

6

Start Page

2166

Last Page

2168

DOI

10.1214/aoms/1177696723

Abstract

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) = ∞.

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Date Posted: 27 November 2017