Riemann 𝑅₁-summability of independent, identically distributed random variables

Cuzick, Jack

doi:10.1090/S0002-9939-1981-0619995-4

Riemann $R_{1}$-summability of independent, identically distributed random variables
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by Jack Cuzick PDF

Proc. Amer. Math. Soc. 83 (1981), 119-124 Request permission

Abstract:

Let $X,{X_1},{X_2}, \ldots$ be i.i.d. random variables. It is shown that $E\left | X \right |{\log ^ + }{\log ^ + }\left | X \right | < \infty$ is a sufficient condition for Riemann ${R_1}$-summability of $\left \{ {{X_n}} \right \}$ to $EX$. Counterexamples are provided which indicate that the strongest possible necessary condition of moment type is $E\left | X \right | < \infty$. However under weak regularity conditions on the tails of the distribution of $X$ the sufficient condition is also shown to be necessary.

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