Compound sums and subexponentiality (Q1962611)

An intuitive interpretation of the subexponential distributions is that a sum of two independent random variables can only exceed a large threshold \(x\) if one of the random variables exceeds the threshold \(x\). A special class of the subexponential distributions is the class of distributions with a regularly varying tail. The compound sums \(S_N=\sum^N_{i=1} Y_i\) are considered, where \(N\) is a random variable taking values in \(\mathbb{N}\) and \((Y_i)\) is a sequence of independent identically distributed random variables independent of \(N\). Under which conditions \(S_N\) or \(N\) belong to the various classes of random variables with the subexponential distribution functions is studied. The proved theorems give a structure of these classes. Some applications can be found in the insurance risk (hurricane insurance) and in a queueing model. An example of the Björk and Grandell process is given.