# Statistics of the longest interval in renewal processes

Research paper by **Claude Godreche, Satya N. Majumdar, Gregory Schehr**

Indexed on: **02 Feb '15**Published on: **02 Feb '15**Published in: **Physics - Statistical Mechanics**

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#### 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.