Abstract
Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\hat{\mathbb{F}}_n$ the empirical distribution function associated with these observations and by $\alpha_n$ the empirical Brownian bridge $\alpha_n(t) = \sqrt n(\hat{\mathbb{F}}_n(t) - t)$, Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge $\mathbb{B}^0$ (depending on $n$) may be constructed on $\Omega$ in such a way that the uniform deviation $\|\alpha_n - \mathbb{B}^0\|_\infty$ between $\alpha_n$ and $\mathbb{B}^0$ is of order of $\log(n)/\sqrt n$ in probability. In this paper, we prove that a Poisson bridge $\mathbb{L}^0_n$ may be constructed on $\Omega$ (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes $\alpha_n, \mathbb{L}^0_n$ and $\mathbb{B}^0$ are of order of $\log(n)/\sqrt n$ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.
Citation
J. Bretagnolle. P. Massart. "Hungarian Constructions from the Nonasymptotic Viewpoint." Ann. Probab. 17 (1) 239 - 256, January, 1989. https://doi.org/10.1214/aop/1176991506
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