  We study the heat statistics of a multi-level $N$-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 $M$ of measurements $(M
\to \infty)$. In this context, the conditions allowing for an
Infinite-Temperature Thermalization (ITT), induced by the repeated 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, to which we refer to as partial thermalization, take place when the
observable of the intermediate measurements is commuting (or quasi-commuting)
with the Hamiltonian of the quantum system, or when the time interval between
measurements is smaller or comparable with the system energy scale (quantum
Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces ($N
\to \infty$), describing continuous systems with a discrete spectrum, are also
presented. We show that the order of the limits $M\to\infty$ and $N\to\infty$
matters: when $N$ is fixed and $M$ diverges, then ITT occurs. 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 instead partial ITT occurs. Finally, an example of partial
thermalization applicable to rotating two-dimensional gases is presented.
