Sample complexity for learning Boltzmann Distribution parameters
Under appropriate smoothness conditions (satisfied by the Boltzmann distribution), the maximum likelihood is asymptotically normal. Meaning: the vector $\sqrt n(\theta-\hat\theta)$ will have approximately normal distribution, with mean $0$ and covariance matrix given by the inverse Fisher matrix, see Section 9 here:
http://www.stat.cmu.edu/~larry/=stat705/Lecture9.pdf