  If we change the upper and lower density in the definition of distributional
chaos of a continuous linear operator on Banach space by the Banach upper and
Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this
fact, we introduce the notions of reiterative distributional chaos of types
$1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are
characterized in terms of the existence of an irregular vector with additional
properties. Moreover, we study its relations with other dynamical properties
and give conditions for the existence of a vector subspace $Y$ of $X$ such that
every non-zero vector in $Y$ is both irregular for $T$ and distributionally
near to zero for $T.$
