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 ${\ell }_{\max }^{\alpha }(t)$ 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 ${\ell }_{\max }^{\alpha }(t)$ 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 ${\ell }_{\max }^{\alpha }(t)/t$ 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.

arXiv comment30 pages, 9 figures, minor revisions

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