On necessary and sufficient conditions for the convergence of the renewal density
Walter L. Smith · Transactions of the American Mathematical Society · 1962
1. Introduction.If k(x) is any non-negative function in Li(-°°, + °°), and if we write k*l(x) = k(x) andn=l is defined almost everywhere (although it is possibly infinite for some, or even almost all, x).Suppose f(x) is the probability density function of a random variable X, the mean value of which ¿11 = 8X1, is strictly positive (it may have the value + + °°, Prior to the important paper of Feller [3] there appears to have been some controversy concerning this asymptotic behavior.Feller provided sufficient conditions under which the renewal density would converge to the desired limit.These conditions have been modified and simplified by Täcklind [10] and by Smith [6;7].The simplest sufficient conditions to date are those given by Smith [7 ] ; these are : (i) f(x) ->0 as x--> + °o ; (¡i) f(x) £ ¿i+í for some 5>0.In the course of some work on the theory of dams we have recently discovered that if /> 00 g-(t+x)xt-1 -^-dt> o r(/) = o,