  All the available results on the approximation of the k-spacings process to
Gaussian processes have only used one approach, that is the Shorack and Pyke's
one. Here, it is shown that this approach cannot yield a rate better than $%
\left( N/\log \log N\right) ^{-\frac{1}{4}}\left( \log N\right) ^{\frac{1}{2}%
}$. Strong and weak bounds for that rate are specified both where k is fixed
and where $k\rightarrow +\infty $. A Glivenko-Cantelli Theorem is given while
Stute's result for the increments of the empirical process based on independent
and indentically distributed random variables is extended to the spacings
process. One of the Mason-Wellner-Shorack cases is also obtained.
