  In this note we study the conditions for convergence of recently introduced
dynamic regressor extension and mixing (DREM) parameter estimator when the
extended regressor is generated using LTI filters. In particular, we are
interested in relating these conditions with the ones required for convergence
of the classical gradient (or least squares), namely the well-known persistent
excitation (PE) requirement on the original regressor vector, $\phi(t) \in
\mathbb{R}^q$, with $q \in \mathbb{N}$ the number of unknown parameters.
Moreover, we study the case when only interval excitation (IE) is available,
under which DREM, concurrent and composite learning schemes ensure global
convergence, being the convergence for DREM in finite time. Regarding PE we
prove that, under some mild technical assumptions, if $\phi(t)$ is PE then the
scalar regressor of DREM, $\Delta(t) \in \mathbb{R}$, is also PE, ensuring
exponential convergence. Concerning IE we prove that if $\phi(t)$ is IE then
$\Delta(t)$ is also IE. All these results are established in the almost sure
sense, namely proving that the set of filter parameters for which the claims do
not hold is of zero measure. The main technical tool used in our proof is
inspired by a study of Luenberger observers for nonautonomous nonlinear systems
recently reported in the literature.
