Statistics of the longest interval in renewal processes. (18th March 2015)

• Record Type: Journal Article
• Authors: Godrèche, Claude; Majumdar, Satya N; Schehr, Grégory
• Is Part Of: Journal of statistical mechanics. (2015:Mar.)
• Publication Date: 2015-03-18
• Subjects: 9; 9/180 -- 9/300 -- 9/165; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/2015/03/P03014
• Languages: English
• ISSNs: 1742-5468

Abstract

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, 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 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/ τ 2 for large τ, then the full statistics of 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 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.