On the Waiting Time Paradox and Related Topics Thierry HUILLET Laboratoire de Physique Th´ eorique et Mod´ elisation, CNRS-ESA8089, Universit´ e de Cergy-Pontoise, 5 mail Gay-Lussac, 95031, Neuville sur Oise, FRANCE fax: (33) 1 34 25 70 04 , e-mail: huillet@ptm.u-cergy.fr Abstract Consider a pure recurrent positive renewal process generated by some in- terarrival waiting time. The waiting time paradox reveals that, asymptotically, the time interval covering one’s arrival in the file is statistically longer than the typical waiting time. Special properties are known to hold, were the waiting time to be infinitely divisible, two particular subclasses of interest being the exponential power mixtures’ and the L´ evy’s ones. These models are revisited in some detail. Questions related to these problems are investigated and special examples of interest are underlined. In press for FRACTALS (March 2002). 1 Renewal processes: definitions and notations Renewal processes (or continuous time random walks) aim at understanding the deep statistical structure of sequences of events occurring randomly in time. We first briefly review part of classical material on this topic [9], [6], [12], [15]. The purpose is to introduce both definitions and notations which are of some use to handle the waiting time paradox problem. 1.1 The elementary counting process Suppose at time t = 0, some event occurs for the first time. Suppose successive events occur in the future in such a way that the waiting times between consecutive events form an independent and identically distributed (iid) sequence (T,T m ,m ≥ 1), with T m d = T , m ≥ 1 (1) We shall need the probability distribution function (pdf) of T and its complement to one (cpdf), i.e. F T (t) := P (T ≤ t), F T (t) := 1 − F T (t) . (2) We shall assume that waiting time T is non-lattice and when T admits a density function (df), we shall call it f T (t). We are then left with a sequence of events 1