Fisher information for exponential family: Regularity conditions for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus.edu.sg/~stachenz/40-530Statistics-6.pdf).

I know that any exponential family satisifies these conditions. But how does one formally show property (iv) on slide 2 in the last link for exponential families, i.e. that one can interchange derivative and integral/expectation? Should one use the dominated convergence theorem? And how does one prove the second assumption (ii) on slide 5 of the link? If you know literature containing the proof I am interested in reading it.

Looking forward to your answers!

Thanks!
I think that this follows from Theorem 5.8 in Theory of Point Estimation by Lehmann and Casella (1998, p.27):


  


you can choose $\eta_i = \theta_i$ (the parameters of a distribution) in the abovementioned formula.
To show both of the properties hold, you can use the Leibniz Integral Rule which lets you pull the derivative inside the integral. Since, for the exponential family $p(x,\theta)$, the partial derivatives of $p(x,\theta)$ with respect to t exist, and is continuous.