  We prove theorems about the Gaussian asymptotics of an empirical bridge built
from linear model regressors with multiple regressor ordering. We study the
testing of the hypothesis of a linear model for the components of a random
vector: one of the components is a linear combination of the others up to an
error that does not depend on the other components of the random vector. The
results of observations of independent copies of a random vector are
sequentially ordered in ascending order of several of its components. The
result is a sequence of vectors of higher dimension, consisting of induced
order statistics (concomitants) corresponding to different orderings. For this
sequence of vectors, without the assumption of a linear model for the
components, we prove a lemma of weak convergence of the distributions of an
appropriately centered and normalized process to a centered Gaussian process
with almost surely continuous trajectories. Assuming a linear relationship of
the components, standard least squares estimates are used to compute regression
residuals, the difference between response values and the predicted ones by the
linear model. We prove a theorem of weak convergence of the process of
regression residuals under the necessary normalization to a centered Gaussian
process. Then we prove a theorem of the same convergence for the empirical
bridge, a self-centered and self-normalized process of regression residuals.
