Limit Forms of the Distribution of the Number of Renewals

In this work the asymptotic properties of $Q_t(N)$ ,the probability of the number of renewals ($N$), that occur during time $t$ are explored. While the forms of the distribution at very long times, i.e. $t\to\infty$, are very well known and are related to the Gaussian Central Limit Theorem or the L\'{e}vy stable laws, the alternative limit of large number of renewals, i.e. $N\to\infty$, is much less noted. We address this limit of large $N$ and find that it attains a universal form that solely depends on the analytic properties of the distribution of renewal times. Explicit formulas for $Q_t(N)$ are provided, together with corrections for finite $N$ and the necessary conditions for convergence to the universal asymptotic limit. Our results show that the Large Deviations rate function for $N/t$ exists and attains an universal linear growth (up to logarithmic corrections) in the $N/t\to\infty$ limit. This result holds irrespective of the existence of mean renewal time or presence of power-law statistics.