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Asymptotic Properties of Convolution Products of Functions
Publications de l'Institut Mathématique, _N_S_43 (1988) no. 57, p. 41
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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^\infty g(y) dy - g(x)\int_0^\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^\infty yg(y) dy+(g(x-1) -g(x))\int_0^\infty yf(y) dy + o(m(x)). $ An application in probability theory is included.
Classification : 27A12
Keywords: convolutions, asymtotic behaviour, subexponential functions, regular variation 
@article{PIM_1988_N_S_43_57_a5,
     author = {Edward Omey},
     title = {Asymptotic {Properties} of {Convolution} {Products} of {Functions}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {41 },
     year = {1988},
     volume = {_N_S_43},
     number = {57},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1988_N_S_43_57_a5/}
}
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Edward Omey. Asymptotic Properties of Convolution Products of Functions. Publications de l'Institut Mathématique, _N_S_43 (1988) no. 57, p. 41 . http://geodesic.mathdoc.fr/item/PIM_1988_N_S_43_57_a5/