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, τ₁, τ₂, …, 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 ${\ell }_{max}^{\alpha }\left(t\right)$ and the probability Q^α(t) that the last interval is the longest one. We show that if ρ(τ) admits a well defined first moment, i.e. if it decays faster than 1/τ² for large τ, then the full statistics of ${\ell }_{max}^{\alpha }\left(t\right)$ 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) ∼ τ^−1−θ for large τ, with index 0 < θ < 1 (like the zero-crossings of random walks corresponding to θ = 1/2), we show that the fluctuations of ${\ell }_{max}^{\alpha }\left(t\right)/t$ are governed, in the large t limit, by a stationary non-trivial universal distribution (different from a Fréchet law) 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 non-trivial constants (depending also on both θ and α) which we compute exactly. (paper)