  For a given graph $G$ without isolated vertex we consider a function $f: V(G)
\rightarrow \{0,1,2\}$. For every $i\in \{0,1,2\}$, let $V_i=\{v\in V(G):\;
f(v)=i\}$. The function $f$ is known to be an outer-independent total Roman
dominating function for the graph $G$ if it is satisfied that; (i) every vertex
in $V_0$ is adjacent to at least one vertex in $V_2$; (ii) $V_0$ is an
independent set; and (iii) the subgraph induced by $V_1\cup V_2$ has no
isolated vertex. The minimum possible weight $\omega(f)=\sum_{v\in V(G)}f(v)$
among all outer-independent total Roman dominating functions for $G$ is called
the outer-independent total Roman domination number of $G$. In this article we
obtain new tight bounds for this parameter that improve some well-known
results. Such bounds can also be seen as relationships between this parameter
and several other classical parameters in graph theory like the domination,
total domination, Roman domination, independence, and vertex cover numbers. In
addition, we compute the outer-independent total Roman domination number of
Sierpi\'nski graphs, circulant graphs, and the Cartesian and direct products of
complete graphs.
