Brownian motion, bridge excursion, and meander characterized by sampling at independent uniform times. (English) Zbl 0935.60068

Pitman, Jim

Electron. J. Probab. 4, Paper No. 11, 33 p. (1999).

The Brownian process \(B\) and processes, derived from this process, such as Brownian bridge, excursion and meander on the interval \([0,1]\), are considered. Let \(X\) be such a process and \(X_{(n)}= (X(\mu_{n,i})\), \(X(U_{n, i});\) \(1\leq i\leq n+1)\), where \((U_{n,1},\dots,U_{n,n})\) are the order statistics of \(n\) independent \((0,1)\)-uniform variables, independent of \(X\); \(\mu_{n,i}\) is the time when \(X\) attains its minimum on the interval \([U_{n,i-1},U_{n,i}]\). It is proved that such a vector coincides in distribution with some other random vector. Components of this vector are composed with sums of independent random variables. For example, if \(X=B\), \[ X_{(n)}{\overset{d}=}\sqrt {2\Gamma_{n+3/2}}\left({S_{i-1} -T_i\over S_{n+1}+T_{n+1}},{S_i-T_i\over S_{n+1} +T_{n+1}}; 1\leq i\leq n+1\right), \] where \(S_n\) is a sum of \(n\) independent standard exponential variables, \((T_i\), \(1\leq i\leq n+1)\) is an independent copy of \((S_i\), \(1\leq i\leq n+1)\) and \(\Gamma_r\) \((r>0)\) is a random variable distributed with \(\Gamma(r)\)-density, independent of \(S_i\) and \(T_i\). Some numerous corollaries are derived from this theorem which partly represent the known results on Brownian process, bridge, excursion and meander.

Reviewer: B.P.Harlamov (St.Peterburg)

MSC:

60J65	Brownian motion
60G50	Sums of independent random variables; random walks

Keywords:

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