  This article describes a new Monte Carlo method for the evaluation of the
orthant probabilities by sampling first passage times of a non-singular
Gaussian discrete time-series across an absorbing boundary. This procedure
makes use of a simulation of several time-series sample paths, aiming to record
their first crossing instants. Thus, the computation of the orthant
probabilities is traced back to the accurate simulation of a non-singular
Gaussian discrete-time series. Moreover, if the simulation is also efficient,
this method is shown to be more speedy than the others proposed in the
literature. As example, we make use of the Davies-Harte algorithm in the
evaluation of the orthant probabilities associated to the ARFIMA$(0,d,0)$
model. Test results are presented that compare this method with currently
available software.
