  We derive exact computable expressions for the asymptotic distribution of the
change-point mle when a change in the mean occurred at an unknown point of a
sequence of time-ordered independent Gaussian random variables. The derivation,
which assumes that nuisance parameters such as the amount of change and
variance are known, is based on ladder heights of Gaussian random walks hitting
the half-line. We then show that the exact distribution easily extends to the
distribution of the change-point mle when a change occurs in the mean vector of
a multivariate Gaussian process. We perform simulations to examine the accuracy
of the derived distribution when nuisance parameters have to be estimated as
well as robustness of the derived distribution to deviations from Gaussianity.
Through simulations, we also compare it with the well-known conditional
distribution of the mle, which may be interpreted as a Bayesian solution to the
change-point problem. Finally, we apply the derived methodology to monthly
averages of water discharges of the Nacetinsky creek, Germany.
