  The unavoidable interaction between a quantum system and the external noisy
environment can be mimicked by a sequence of stochastic measurements whose
outcomes are neglected. Here we investigate how this stochasticity is reflected
in the survival probability to find the system in a given Hilbert subspace at
the end of the dynamical evolution. In particular, we analytically study the
distinguishability of two different stochastic measurement sequences in terms
of a new Fisher information measure depending on the variation of a function,
instead of a finite set of parameters. We find a novel characterization of Zeno
phenomena as the physical result of the random observation of the quantum
system, linked to the sensitivity of the survival probability with respect to
an arbitrary small perturbation of the measurement stochasticity. Finally, the
implications on the Cram\'er-Rao bound are discussed, together with a numerical
example. These results are expected to provide promising applications in
quantum metrology towards future, more robust, noise-based quantum sensing
devices.
