Statistics of the longest interval in renewal processes

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, $τ_{1},τ_{2},...$, are independent and identically distributed (i.i.d.) random variables with a common density $ρ(τ)$. 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 $α=$ I, II and III. We investigate the distribution of the longest interval $\ell^α_{\max}(t)$ and the probability $Q^α(t)$ that the last interval is the longest one. We show that if $ρ(τ)$ decays faster than $1/τ^2$ for large $τ$, then the full statistics of $\ell^α_{\max}(t)$ 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 $τ_i$ does not play any role at large times in this case. However, if $ρ(τ)$ exhibits heavy tails, $ρ(τ)\simτ^{-1-θ}$ for large $τ$, with index $0 <θ<1$, we show that the fluctuations of $\ell^α_{\max}(t)/t$ are governed, in the large $t$ limit, by a stationary universal distribution which depends on both $θ$ and $α$, which we compute exactly. On the other hand, $Q^α(t)$ is generically different from its counterpart for i.i.d. variables (both for narrow or heavy tailed distributions $ρ(τ)$). In particular, in the case $0<θ<1$, the large $t$ behaviour of $Q^α(t)$ gives rise to universal constants (depending also on both $θ$ and $α$) which we compute exactly.

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