On the maximum of randomly weighted sums with subexponential tails

Consider the randomly weighted sums Sn(µ) = P n=1 µkXk, where fXk, 1 • kng is a sequence of independent real-valued random variables with common subexponential distribution function F, and let fµk;1 • kng a sequence of positive random variables, independent of fXk;1 • kng and satisfying a • µkb for some 0 < ab < 1 for all 1 • kn. Under a suitable summability condition on the upper endpoints of jµkj we prove that