  We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic
moduli depend on $n$ separated microscopic scales, are random and expressed as
a linear expansion of a countable sequence of random variables which are
independently and identically uniformly distributed in a compact interval. The
multiscale Hellinger-Reissner problem that allows for computing the stress
directly, and the multiscale mixed problem for nearly incompressible isotropic
materials are considered. The stochastic problems are studied via deterministic
problems that depend on a countable number of real parameters. We study the
multiscale homogenized problems that contain all the macroscopic and
microscopic information, whose solutions are written as generalized polynomial
chaos (gpc) expansions. We approximate these solutions by semidiscrete Galerkin
approximating problems that project into the spaces of functions with only a
finite number of $N$ gpc modes. We deduce bounds and summability properties for
the solutions' gpc expansion coefficients, which imply explicit rates of
convergence in terms of $N$ when the gpc modes used for the Galerkin
approximation are chosen to correspond to the best $N$ terms in the gpc
expansion. For the mixed problem for nearly incompressible materials, the rate
of convergence for the best $N$ term approximation is independent of the Lam\'e
constants' ratio. We establish parametric correctors in terms of the
semidiscrete Galerkin approximations. For two scale problems, an explicit
homogenization rate is deduced. Together with the best $N$ term rate, it
provides an explicit convergence rate for the correctors of the parametric
multiscale problems. For nearly incompressible materials, we obtain a
homogenization rate that is independent of the ratio of the Lam\'e constants,
so that the error for the corrector is also independent of this ratio.
