Exponential loss of memory for random Lasota--Yorke convex maps with local average expansion

We study i.i.d. random compositions of piecewise convex interval maps whose first branch is full. Uniform expansion is not required, and the maps may have contracting branches. Under a local average expansion condition at the origin, we prove both quenched and annealed exponential loss of memory for initial densities of bounded variation. The proof is based on a coupling argument. As applications, we obtain the quenched almost sure invariance principle and a rate of convergence in the quenched central limit theorem.