  We introduce a new technique to establish Hungarian multivariate theorems. In
this article we apply this technique to the strong approximation bivariate
theorems of the uniform empirical process. It improves the Komlos, Major and
Tusn\'ady (1975) result, as well as our own (1998). More precisely, we show
that the error in the approximation of the uniform bivariate $n$-empirical
process by a bivariate Brownian bridge is of order $n^{-1/2}(log (nab))^{3/2}$
on the rectangle $[0,a]x[0,b]$, $0 <a, b <1$, and that the error in the
approximation of the uniform univariate $n$-empirical process by a Kiefer
process is of order $n^{-1/2}(log (na))^{3/2}$ on the interval $[0,a]$, $0 < a
< 1$. In both cases, the global error bound is therefore of order $n^{-1/2}(log
(n))^{3/2}$. Previously, from the 1975 article of Komlos, Major and Tusn\'ady,
the global error bound was of order $n^{-1/2}(log (n))^{2}$, and from our 1998
article, the local error bounds were of order $n^{-1/2}(log (nab))^{2}$ or
$n^{-1/2}(log (na))^{2}$. We think that, in the d-variate case, the global
error bound between the uniform d-variate $n$-empirical process and the
associated Gaussian process is of order $n^{-1/2}(log (n))^{(d+1)/2}$, and that
this result is optimal. The new feature of this article is to identify
martingales in the error terms and to apply to them an exponential inequality.
The idea is to bound of the compensator of the error term, instead of bounding
of the error term itself.
