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

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.

### MSC:

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