  Recently found positive representation for an arbitrary complex, gaussian
weight is used to construct a statistical formulation of gaussian path
integrals directly in the Minkowski time. The positivity of Minkowski weights
is achieved by doubling the number of real variables. The continuum limit of
the new representation exists only if some of the additional couplings tend to
infinity and are tuned in a specific way. The construction is then successfully
applied to three quantum mechanical examples including a particle in a constant
magnetic field -- a simplest prototype of a Wilson line. Further
generalizations are shortly discussed and an intriguing interpretation of new
variables is alluded to.
