Statistics of the longest interval in renewal processes

Abstract

We consider renewal processes where events, which can for instance be the zero crossings of a stochastic process, occur at random epochs of time. The intervals of time between events, $\tau_{1},\tau_{2},...$, are independent and identically distributed (i.i.d.) random variables with a common density $\rho(\tau)$. Fixing the total observation time to $t$ induces a global constraint on the sum of these random intervals, which accordingly become interdependent. Here we focus on the largest interval among such a sequence on the fixed time interval $(0,t)$. Depending on how the last interval is treated, we consider three different situations, indexed by $\alpha=$ I, II and III. We investigate the distribution of the longest interval $\ell^\alpha_{\max}(t)$ and the probability $Q^\alpha(t)$ that the last interval is the longest one. We show that if $\rho(\tau)$ decays faster than $1/\tau^2$ for large $\tau$, then the full statistics of $\ell^\alpha_{\max}(t)$ is given, in the large $t$ limit, by the standard theory of extreme value statistics for i.i.d. random variables, showing in particular that the global constraint on the intervals $\tau_i$ does not play any role at large times in this case. However, if $\rho(\tau)$ exhibits heavy tails, $\rho(\tau)\sim\tau^{-1-\theta}$ for large $\tau$, with index $0 <\theta<1$, we show that the fluctuations of $\ell^\alpha_{\max}(t)/t$ are governed, in the large $t$ limit, by a stationary universal distribution which depends on both $\theta$ and $\alpha$, which we compute exactly. On the other hand, $Q^{\alpha}(t)$ is generically different from its counterpart for i.i.d. variables (both for narrow or heavy tailed distributions $\rho(\tau)$). In particular, in the case $0<\theta<1$, the large $t$ behaviour of $Q^\alpha(t)$ gives rise to universal constants (depending also on both $\theta$ and $\alpha$) which we compute exactly.