# pr.probability – Regularity of pdf of partial sums of Birkhoff

Assume that $$T: X to X$$ is a measurable map on a Riemannian variety $$X$$ (possibly with limit). Let $$mu$$ denote the Riemannian measure on $$X$$. For measurable, real value $$g$$ we can consider the partial sums of Birkhoff $$S_n: = sum_ {k = 0} ^ {n-1} g circ T ^ k$$.
I am particularly interested in the case where $$T$$ has some hyperbolicity, for example $$T$$ could be a piece $$mathcal {C} ^ 2$$ map expanding on $$[0,1]$$, or one $$mathcal {C} ^ 2$$ Anosov map on $$mathbb {T} ^ 2$$. Also, $$g$$ can be as regular as necessary.

Yes $$g$$ and $$T$$ to have enough regularity, then $$S_n$$ has a probability density function $$f_n$$. My question is whether anyone has studied the regularity of $$f_n$$: in what conditions $$f_n$$ exists, and what can we say about it $$n to infty$$. If we define the Fischer information of $$S_n$$ be
$$I (S_n) = int _ { mathbb {R}} left[frac{mathrm{d}}{mathrm{d}x} log f_nright]^ 2 f_n (x) , mathrm {d} x,$$
can we say if $$I (S_n)$$ is finished? If so, can we say something about the growth of $$I (S_n)$$ as $$n to infty$$?