Strong Renewal Theorems with Infinite Mean

Let F be a nonarithmetic probability distribution on <!-- MATH $(0,\infty )$ --> and suppose is regularly varying at with exponent <!-- MATH $\alpha ,0 < \alpha \leqq 1$ --> <img width="113" height="41" align="MIDDLE" border="0" src="images/img4.gif" alt="$ \alpha ,0 < \alpha \leqq 1$">. Let <!-- MATH $U(t) = \Sigma {F^{{n^ \ast }}}(t)$ --> be the renewal function. In this paper we first derive various asymptotic expressions for the quantity <!-- MATH $U(t + h) - U(t)$ --> as <!-- MATH $t \to \infty ,h > 0$ --> 0$"> fixed. Next we derive asymptotic relations for the convolution <!-- MATH ${U^ \ast }z(t),t \to \infty$ --> , for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function <!-- MATH $m(t) = \smallint _0^t[1 - F(x)]dx$ --> , t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case <!-- MATH $\alpha = 1$ --> to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.