  Consider the stationary measure of open asymmetric simple exclusion process
(ASEP) on the lattice $\{1,\dots,n\}$. Taking $n$ to infinity while fixing the
jump rates, this measure converges to a measure on the semi-infinite lattice.
In the high and low density phases, we characterize the limiting measure and
also show that this convergence occurs in total variation distance on a
sublattice of scale $n/\log n$. Our approach involves bounding the total
variation distance using generating functions, which are further estimated
through a subtle analysis of the atom masses of Askey-Wilson signed measures.
