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We improve on some results of SUNDT (1982) on the asymptotic behavlour of compound negative binomial distributions KEY WORDS Compound negative binomial distributions, renewal theory, asymptotic estimates. Consider the aggregate claims of an insurance company in a given period, N x=EY, where the claim sizes {Y,: ~No} are i.i.d, non-negative random variables with F(x) = P { Y ~ ~< x} non-lattice (i.e., we assume the claim size distribution F to be non-discrete; take for instance F continuous), independent of the negative binomial claim arrwal variable N. Then pn~P{N n}=(a~- I+n) = p"q", n~ n where 0 < p < 1, p + q = I and o ~> 0. Denote by f the Laplace-Stieltjes transform of F and assume that there exists a constant K> 0 satisfying;o o p-i = e ~ dF(x) (I) and that (2) v=p xe'X dF(x)<oo i.e, p- ' =f (-K) and If '(-K)l <oo. We now want to estimate P{X>x} as x~co