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For a sequence of immersed connected closed Hamiltonian stationary Lagrangiansubmaniolds in $\mathbb{C}^{n}$ with uniform bounds on their volumes and thetotal extrinsic curvatures, we prove that a subsequence converges either to apoint or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformlyin $C^{k}$ for any nonnegative integer $k$ away from a finite set of points,and the limit is Hamiltonian stationary in ${\mathbb{C}}^{n}$. We also obtain atheorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ acrossa compact set $N$ of Hausdorff codimension at least 2 that is locallynoncollapsing in volumes matching its Hausdorff dimension, provided the meancurvature of $L$ is in $L^{n}$ and a condition on local volume of $L$ near $N$is satisfied.

Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume