Electronic Journal of Probability

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

Jim Pitman

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For a random process $X$ consider the random vector defined by the values of $X$ at times $0 \lt U_{n,1} \lt ... \lt U_{n,n} \lt 1$ 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.

Article information

Electron. J. Probab., Volume 4 (1999), paper no. 11, 33 pp.

Accepted: 26 April 1999
First available in Project Euclid: 4 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 05A19: Combinatorial identities, bijective combinatorics 11B73: Bell and Stirling numbers

alternating exponential random walk uniform order statistics critical binary random tree Vervaat's transformation random partitions generalized Stirling numbers Bessel polynomials McDonald function products of gamma variables Hermite function

This work is licensed under aCreative Commons Attribution 3.0 License.


Pitman, Jim. Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times. Electron. J. Probab. 4 (1999), paper no. 11, 33 pp. doi:10.1214/EJP.v4-48. https://projecteuclid.org/euclid.ejp/1457125520

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