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An asymptotic series expansion of the multidimensional renewal measure

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An asymptotic series expansion of the multidimensional renewal measure COMPOSITIO MATHEMATICA HASSECARLSSON STEPHENWAINGER An asymptotic series expansion of the multidimensional renewalmeasure Compositio Mathematica, tome 47, no 3 (1982), p. 355-364. <> © Foundation Compositio Mathematica, 1982, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques 355 AN ASYMPTOTIC SERIES EXPANSION OF THE MULTIDIMENSIONAL RENEWAL MEASURE Hasse Carlsson1 and Stephen Wainger2 COMPOSITIO MATHEMATICA, Vol. 47, Fasc. 3, 1982, pag. 355-364 © 1982 Martinus Nijhoff Publishers - The Hague Printed in the Netherlands 1. Introduction and main theorem Let p(x) dx be an absolutely continuous probability distribution on Euclidean d-dimensional space with non-vanishing mean vector 03BC. As usual we define the renewal measure v by the formula for any Borel set E. Then it is well known that, if say Q is the unit cube, where p = 1 2(d - 1). See [1], [2], [4], and [6]. We are concerned here with the error In one dimension the decay of E(03BB) is to a large extent independent of the distribution p(x), provided p(x) has sufficiently many moments. For example if 11 |x|kp(x)dx + oo, k = 2, 3, 4,..., then 1 Supported by the Swedish Natural Science Research Council. 2 Supported in part by an N.S.F. grant at the University of Wisconsin. 0010-437X/82090355-10$00.20/0 356 (See however Stone [7] where it is shown that where R(03BB) does depend on p(x).) In contrast to this, in

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