Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums S-n = Sigma(i=1)(n) X-i, ngreater than or equal to1; S(t) = Sigma(i=1)(N)(t) X-i, tgreater than or equal to0, where {N(t); tgreater than or equal to0} are non-negative integer-valued random variables and {X-n; ngreater than or equal to1} are independent non-negative random variables with distribution, F-n, of X-n, independent of {N(t); tgreater than or equal to0}. Special attention is paid to the distribution of dominated variation.