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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. <http://www.numdam.org/item?id=CM_1982__47_3_355_0> © Foundation Compositio Mathematica, 1982, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). 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 http://www.numdam.org/ 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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