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

Bretagnolle, J.; Massart, P.

Ann. Probab. 17, No. 1, 239-256 (1989).

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

Keywords:

strong approximation; empirical process; Brownian bridges; Poisson process

Citations:

Zbl 0308.60029