I am working on the Fisher information of a density $\rho$, defined as: $I(\rho) = \int \left\lVert \nabla_{x} \log \rho(x) \right\rVert^2 \rho(x) \mathrm{d} x$. The gradient is taken with respect to $x$. I couldn't find much resources on this integral. Can anyone suggest some notes for me?
I have two random variables with densities: $X \sim \rho_0(x), Y \sim \rho_1(y)$. I am studying the sum of these two variables: $Z = X + Y$ or $Z = aX + bY$ for parameters $a, b$. $Z$ will follow another density $p(z)$. What is the relationship between the Fisher information of these three variables? That is, what is the relationship between $I(p), I(\rho_0), I(\rho_1)$.
If it is possible to derive a property about $I(p)$, it would also be great.
Thank you.