  The behaviour of a stochastic dynamical system may be largely influenced by
those low-probability, yet extreme events. To address such occurrences, this
paper proposes an infinite-horizon risk-constrained Linear Quadratic Regulator
(LQR) framework with time-average cost. In addition to the standard LQR
objective, the average one-stage predictive variance of the state penalty is
constrained to lie within a user-specified level. By leveraging the duality,
its optimal solution is first shown to be stationary and affine in the state,
i.e., $u(x,\lambda^*) = -K(\lambda^*)x + l(\lambda^*)$, where $\lambda^*$ is an
optimal multiplier, used to address the risk constraint. Then, we establish the
stability of the resulting closed-loop system. Furthermore, we propose a
primal-dual method with sublinear convergence rate to find an optimal policy
$u(x,\lambda^*)$. Finally, a numerical example is provided to demonstrate the
effectiveness of the proposed framework and the primal-dual method.
