Asymptotic Properties of Convolution Products of Functions

Edward Omey

Publications de l'Institut Mathématique, _N_S_43 (1988) no. 57, p. 41 .

Résumé

The asymptotic behaviour of convolution products of the form ${\int }_0^{x}f(x-y)g(y)dy$ is studied. From our results we obtain asymptotic expansions of the form $R(x):={\int }_o^{x}f(x-y)g(y)dy-f(x){\int }^{\mathrm{\infty }}g(y)dy-g(x){\int }_0^{\mathrm{\infty }}f(y)dy=O(m(x)).$ Under rather mild conditions on $f,g$ and $m$ the $O$-term can be calculated more explicitly as $R(x)-(f(x-1)-f(x)){\int }_0^{\mathrm{\infty }}yg(y)dy+(g(x-1)-g(x)){\int }_0^{\mathrm{\infty }}yf(y)dy+o(m(x)).$ An application in probability theory is included.

Détail

Classification : 27A12
Mots-clés : convolutions, asymtotic behaviour, subexponential functions, regular variation