  We study the heat statistics of an $N$-dimensional quantum system monitored
by a sequence of projective measurements. The late-time, asymptotic properties
of the heat characteristic function are analyzed in the thermodynamic limit of
a high, ideally infinite, number of measurements. In this context, conditions
allowing for Infinite-Temperature Thermalization (ITT), induced by the
monitoring of the quantum system, are discussed. We show that ITT is identified
by the fixed point of a symmetric random matrix that models the stochastic
process originated by the sequence of measurements. Such fixed point is
independent on the non-equilibrium evolution of the system and its initial
state. Exceptions to ITT take place when the observable of the intermediate
measurements is commuting (or quasi-commuting) with the Hamiltonian of the
system, or when the time interval between measurements is smaller or comparable
with the energy scale of the quantum system (quantum Zeno regime). Results on
the limit of infinite-dimensional Hilbert spaces ($N \to \infty$) - describing
continuous systems with a discrete spectrum - are presented. By denoting with
$M$ the number of quantum measurements, we show that the order of the limits $M
\to \infty$ and $N \to \infty$ matters: when $N$ is fixed and $M$ diverges,
then there is ITT. In the opposite case, the system becomes classical, so that
the measurements are no longer effective in changing the state of the system. A
non trivial result is obtained fixing $M/N^2$ where partial ITT occurs.
Finally, an example of incomplete thermalization applicable to rotating
two-dimensional gases is presented.
