Statistics of the longest interval in renewal processes

TL;DR

This paper analyzes the distribution of the longest interval in renewal processes with fixed total time, revealing universal behaviors depending on the tail properties of the interval distribution.

Contribution

It provides exact results for the distribution of the longest interval and the probability of it being the last, considering different tail behaviors of the interval distribution.

Findings

For heavy-tailed distributions, the longest interval's fluctuations follow a universal distribution.

The probability that the last interval is the longest converges to universal constants depending on tail index.

Standard extreme value theory applies for distributions decaying faster than 1/τ^2.

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.