Geodesic
Semimultiplicative estimates for the solution of the multidimensional renewal equation
Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 595-610.

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Expansions with semimultiplicative estimates for the remainders are found for the solutions of the multidimensional renewal equation. The effect of the roots of the characteristic equation on the asymptotic expression for the solution is taken into consideration. These results are used to investigate the asymptotic behaviour of the average number of particles in relation to the age of the branching processes in several cases. 
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M. S. Sgibnev. Semimultiplicative estimates for the solution of the multidimensional renewal equation. Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 595-610. http://geodesic.mathdoc.fr/item/IM2_2002_66_3_a5/

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