Moderate deviations for random sums of heavy-tailed random variables

Let {X-n; n >= 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean mu = E(X-1) and let {N(t); t = 0} be a random process taking non-negative integer values with finite mean lambda(t) = E( N(t)) and independent of {Xn; n = 1}. In this paper, asymptotic expressions of P((X-1+center dot center dot center dot+ X-N(T))-lambda(t)mu > x) uniformly for x epsilon [gamma b(t),infinity) are obtained, where gamma > 0 and b(t) can be taken to be a positive function with lim(t ->infinity)b(t)/lambda(t) = 0.