The construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension. This paper deals with the effective construction of the probability distribution in high dimension of a vector-valued random variable using the maximum entropy principle. The integrals in high dimension are then calculated in constructing the stationary solution of an lto stochastic differential equation associated with its invariant measure. A random generator of independent realizations is explicitly constructed in this paper. Three fundamental applications are presented. The first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non-stationary random time series (random accelerograms) for which the velocity response spectrum is given. The second one is also a new formulation related to the construction of the probability distribution of positive-definite band random matrices. Finally, we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space. The third application is then a new formulation related to the construction of the probability distribution of the Karhunen-Loeve expansion of non-Gaussian positive-valued random fields.