The rank of the present excursion

It is shown that for standard Brownian motion the present excursion from 0 has probability 0.80031 … of establishing a record as far as its duration is concerned. This follows from a more general result concerning the rank of an excursion in the point process representation of a renewal process having interarrival times, the distribution of which varies regularly at infinity with exponent p ϵ] − 1,0[. Using Palm probabilities for Poisson processes one finds in fact the generating function of this rank explicitly.