  In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve
linear elasticity problem with arbitrary order discontinuous finite element
spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$
for the stress and of degree $k$ for the displacement ($k\geq 0$). The mixed DG
scheme is proved to be well-posed under proper norms. Specifically, we prove
that, for any $k \geq 0$, the $H({\rm div})$-like error estimate for the stress
and $L^2$ error estimate for the displacement are optimal. We further establish
the optimal $L^2$ error estimate for the stress provided that the
$\mathcal{P}_{k+2}-\mathcal{P}_{k+1}^{-1}$ Stokes pair is stable and $k \geq
d$. We also provide numerical results of MDG showing that the orders of
convergence are actually sharp.
