## Hungarian constructions from the nonasymptotic viewpoint.(English)Zbl 0667.60042

Let $$X_ 1,X_ 2,...,X_ n$$ be independent r.v.’s with uniform distribution on [0,1] defined on a rich enough probability space $$\Omega$$. Let $$F_ n(t)$$ be the empirical distribution function based on the sample $$X_ 1,X_ 2,...,X_ n$$ and let $$\alpha_ n(t)=\sqrt{n}(F_ n(t)-t)$$ (0$$\leq t\leq 1)$$ be the empirical process. The authors are interested in defining a sequence of Brownian bridges $$B_ n(t)$$ (0$$\leq t\leq 1)$$ and a sequence of Poisson processes $$P_ n(t)$$ (0$$\leq t\leq \infty)$$ on $$\Omega$$ such that the distances $\sup_{0\leq t\leq 1}| B_ n(t)-\alpha_ n(t)| \quad and\quad \sup_{0\leq t\leq 1}| P_ n^{(0)}(t)-\alpha_ n(t)|$ become as small as possible where $P_ n^{(0)}(t)=n^{-1/2}(P_ n(nt)-P_ n(n))\quad (0\leq t\leq 1).$ The main result states that the sequences $$P_ n(t)$$ and $$B_ n(t)$$ can be constructed such that $P\{n^{1/2}D_ n>x+12 \log n\}\leq 2 \exp (-x/6)$ for any $$x>0$$ where $$D_ n$$ is any of the above two distances. This result is in case of the Brownian bridge a stronger version of the result of J. Komlos, P. Major and G. Tusnady [Z. Wahrscheinlichkeitstheorie verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)] and very new in case of a Poisson process.
Reviewer: P.Révész

### MSC:

 60F17 Functional limit theorems; invariance principles 60F99 Limit theorems in probability theory 62G30 Order statistics; empirical distribution functions

Zbl 0308.60029
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