Abstract. 
In the first half of the talk I will outline results concerning stability 
with respect to various perturbations of the variance in a central limit 
theorem, and the rate function in a large deviation principle. These 
perturbations include those arising from numerical methods and also 
allow us to estimate SRB measures of Anosov maps of the torus. In the 
second half of the talk I will outline an extension of the powerful 
spectral approach to proving limit laws like large deviations 
principles and CLTs to the case of randomly driven dynamics. 
Our approach remains spectral and we prove so-called "quenched" results,
which hold for almost-every initialisation of the driving system.