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We prove large deviation results on the partial and random sums Sn = ∑ni=1 Xi, n≥1; S(t) =∑N(t)i=1 Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} areindependent non-negative random variables with distribution, Fn,