Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

For a random process X consider the random vector deened by the values of X at times 0 < U n;1 < ::: < U n;n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n;i are the order statistics of n independent uniform (0; 1) variables, independent of X. The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simpliied proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other 1 consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained. Contents 1 Introduction 2 2 Summary of Results 4 3 Proofs of the Sampling Formulae 14 4 Bessel polynomials, products of independent gamma variables, and the meander 18 5 Combinatorial identities related to random partitions 23 6 Distributions of some total variations 26 7 Decomposition at the minimum 30 8 Bridges with arbitrary end point 33