Remainder Term Estimates in One and Many-Dimensional Renewal Theory

摘要

Let X1, X2, ... be d-dimensional independent random vectors with a common distribution mu. Let nu = the summation of n = 0 to infinity mu exponent N* and, for d = 1, H(X) = NU (-infinity, X). The asymptotic behavior of NU (A+X) and H(X) as the absolute value of X tends to infinity are investigated. It is known that H(X) - X/mu sub 1 - mu sub 2/2 mu 1 to 2 tends to zero as X tends to + infinity if mu sub 1 zero and mu is a non-lattice measure (mu sub k is the kth moment of mu). The rate of this convergence under further condition on mu is considered. The estimates are obtained by a Fourier transform technique making use of Bohr's inequality. The problem is also treated for d or = Z.