  The aim of this article is to understand the geometry of limit sets in
pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of
$\mathrm{PO}(p,q+1)$ introduced by Danciger, Gu\'eritaud and Kassel, called
$\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of
critical exponent and Hausdorff dimension of the limit set. We show that they
are equal and bounded from above by the usual Hausdorff dimension of the limit
set. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which
can be understood as a Lorentzian version of a famous Theorem of R. Bowen in
$3$-dimensional hyperbolic geometry.
