  A meaningful probability distribution for measurements of a quantum stress
tensor operator can only be obtained if the operator is averaged in time or in
spacetime. This averaging can be regarded as a description of the measurement
process. Realistic measurements can be expected to begin and end at finite
times, which means that they are described by functions with compact support,
which we will also take to be smooth. Here we study the probability
distributions for stress tensor operators averaged with such functions of time,
in the vacuum state of a massless free field. Our primary aim is to understand
the asymptotic form of the distribution which describes the probability of
large vacuum fluctuations. Our approach involves asymptotic estimates for the
high moments of the distribution. These estimates in turn may be used to obtain
estimates for the asymptotic form of the probability distribution. Our results
show that averaging over a finite interval results in a probability
distribution which falls more slowly than for the case of Lorentzian averaging,
and both fall more slowly than exponentially. This indicates that vacuum
fluctuations effects can dominate over thermal fluctuations in some
circumstances.
