  A continuous projective measurement of a quantum system often leads to a
suppression of the dynamics, known as the Zeno effect. Alternatively,
generalized nonprojective, so-called "weak" measurements can be carried out.
Such a measurement is parameterized by its strength parameter that can
interpolate continuously between the ideal strong measurement with no
dynamics-the strict Zeno effect, and a weak measurement characterized by almost
free dynamics but blurry observations. Here we analyze the stochastic
properties of this uncertainty component in the resulting observation
trajectory. The observation uncertainty results from intrinsic quantum
uncertainty, the effect of measurement on the system (backaction) and detector
noise. It is convenient to separate the latter, system-independent contribution
from the system-dependent uncertainty, and this paper shows how to accomplish
this separation. The system-dependent uncertainty is found in terms of a
quasi-probability, which, despite its weaker properties, is shown to satisfy a
weak positivity condition. We discuss the basic properties of this
quasi-probability with special emphasis on its time correlation functions as
well as their relationship to the full correlation functions along the
observation trajectory, and illustrate our general results with simple
examples.We demonstrate a violation of classical macrorealism using the
fourth-order time correlation functions with respect to the quasi-probability
in the twolevel system.
