  In this paper we study the estimation of changing trends in time-series using
$\ell_1$ trend filtering. This method generalizes 1D Total Variation (TV)
denoising for detection of step changes in means to detecting changes in
trends, and it relies on a convex optimization problem for which there are very
efficient numerical algorithms. It is known that TV denoising suffers from the
so-called stair-case effect, which leads to detecting false change points. The
objective of this paper is to show that $\ell_1$ trend filtering also suffers
from a certain stair-case problem. The analysis is based on an interpretation
of the dual variables of the optimization problem in the method as integrated
random walk. We discuss consistency conditions for $\ell_1$ trend filtering,
how to monitor their fulfillment, and how to modify the algorithm to avoid the
stair-case false detection problem.
