I will talk about an almost sure invariance principle in parameter space for the
quadratic family, as inspired by a result of Daniel Schnellmann for piecewise
expanding systems, and obtained together with Viviane Baladi and Magnus Aspenberg.
The almost sure invariance principle implies several statistical properties such as the central limit theorem and the law of the iterated logarithm. The result for the quadratic family therefore says that ergodic averages as a function of the parameter has a normal distribution after appropriate scalings. Hopefully this has an application to fractional response, in the style of a paper by Amanda de Lima and Daniel Smania.