Complete exact laws (Q2772049)

Laws of large numbers for independent and identically distributed random variables \(\{ X,X_n\), \(n\geq 1 \}\) with \(xP\{X>x\}\sim a(\log x)^{\alpha},\) where \(\alpha >-1\) and \(P\{X<-x\}= o(P\{X>x\})\) are considered. Even the mean does not exist, the laws of large numbers of the form NEWLINE\[NEWLINE \sum_{n=1}^{\infty} c_n P\left\{ \left|\frac{\sum_{k=1}^n a_k X_k}{b_n}-L \right|> \varepsilon \right\}<\infty NEWLINE\]NEWLINE for all \(\varepsilon >0\) and a particular nonsummable sequence \(\{c_n\), \(n\geq 1\},\) where \(L\neq 0\), are established.