  Large vacuum fluctuations of a quantum stress tensor operator can be
described by the asymptotic behavior of the probability distribution of the
time or spacetime averaged operator. Here we focus on the case of stress tensor
operators averaged with a sampling function in time. The Minkowski vacuum state
is not an eigenstate of the time-averaged operator, but can be expanded in
terms of its eigenstates. We calculate the probability distribution and the
cumulative probability distribution for obtaining a given value in a
measurement of the time-averaged operator taken in the vacuum state. In these
calculations, we use the normal ordered square of the time derivative of a
massless scalar field in Minkowski spacetime as an example of a stress tensor
operator. We analyze the rate of decrease of the tail of the probability
distribution for different temporal sampling functions, such as compactly
supported functions and the Lorentzian function. We find that the tails
decrease relatively slowly, as exponentials of fractional powers, in agreement
with previous work using the moments of the distribution. Our results lead
additional support to the conclusion that large vacuum stress tensor
fluctuations are more probable than large thermal fluctuations, and may have
observable effects.
