Geodesic
On the explicit estimates for the power rate of convergence in the renewal theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 600-607

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The renewal equation $x(t)=y(t)+\int_0^t x(t-s)\,dF(s)$ is considered. The explicit estimates are obtained for $$ \sup_t(1+\varepsilon t)^{\alpha}|x(t)-\lim_{t\to\infty}x(t)| $$ under the assumptions on the power decay of $y(t)$ and on the existence of moments of $F(t)$ for some classes of distribution functions $F(t)$. One of this classes include distributions having independent exponential component. 
@article{TVP_1979_24_3_a15,
     author = {N. V. Karta\v{s}ov},
     title = {On the explicit estimates for the power rate of convergence in the renewal theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {600--607},
     publisher = {mathdoc},
     volume = {24},
     number = {3},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a15/}
}
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N. V. Kartašov. On the explicit estimates for the power rate of convergence in the renewal theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 600-607. http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a15/