Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times
Jim Pitman, University of California, Berkeley
Abstract
For a random process $X$ consider the random vector defined 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 simplified 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 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.
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