Identically distributed uncorrelated random variables not fulfilling the WLLN
Bull. Korean Math. Soc. 2001 Vol. 38, No. 3, 605-610
Dieter Landers and Lothar Rogge
University of Cologne, Gerhard-Mercator-Universitat Duisburg
Abstract : It is shown that for each $1 < p < 2$ there exist identically
distributed uncorrelated random variables $X_{n}$ with
$E(|X_{1}|^{p}) < \infty,$ not fulfilling the weak law
of large numbers (WLLN). If, however, the random variables
are moreover non-negative, the weaker integrability condition
$E(X_{1}\log X_{1}) < \infty$ already guarantees the strong law of large numbers.
Keywords : weak law of large numbers, strong law of large numbers, uncorrelated identically distributed random variables
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