The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance

In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary real-valued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as S-m((w)), represents the stochastic present value of aggregate net losses during the first m periods. Suppose that the primary random variables are independent of each other with long-tailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of max(1 <= m <= n)S(m)((w))-the maximum of the first n weighted sums-is asymptotically equivalent to that of S-n((w))-the last weighted sum.