Bulletin of the
Korean Mathematical Society BKMS

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