On relative stability and weighted laws of large numbers

Weighted laws of large numbers are established for components which are independent copies of a positive relatively stable law and the weights comprise a regularly varying sequence. The index of regular variation of the weights must be at least −1 for a weak law and be exactly −1 for a strong law. Consideration is given to the special case where the truncated moment function is proportional to the logarithm, a case arising from quotients of independent random variables and quotients of successive order statistics. Closure of the class of positive relatively stable laws under independent multiplication (i.e, Mellin convolution) is reprised and tail equivalences are extended.