  The unpredictability of chaotic nonlinear dynamics leads naturally to
statistical descriptions, including probabilistic limit laws such as the
central limit theorem and large deviation principle. A key tool in the
Nagaev-Guivarc'h spectral method for establishing statistical limit theorems is
a "twisted" transfer operator. In the abstract setting of Keller-Liverani we
prove that derivatives of all orders of the leading eigenvalues and
eigenprojections of the twisted transfer operators with respect to the twist
parameter are stable when subjected to a broad class of perturbations. As a
result, we demonstrate stability of the variance in the central limit theorem
and the rate function from a large deviation principle with respect to
deterministic and stochastic perturbations of the dynamics and perturbations
induced by numerical schemes. We apply these results to piecewise expanding
maps in one and multiple dimensions, including new convergence results for Ulam
projections on quasi-H\"older spaces.
