  The monotone rearrrangement algorithm was introduced by Hardy, Littlewood and
P\'olya as a sorting device for functions. Assuming that $x$ is a monotone
function and that an estimate $x_n$ of $x$ is given, consider the monotone
rearrangement $\hat{x}_n$ of $x_n$. This new estimator is shown to be uniformly
consistent. Under suitable assumptions, pointwise limit distribution results
for $\hat{x}_n$ are obtained. The framework is general and allows for weakly
dependent and long range dependent stationary data. Applications in monotone
density and regression function estimation are detailed.
