Let $H$ be a continuous energy function defined on a compact subset $A\subset \mathbf{R}^n$ and let $Q$ be a fixed probability measure on $A$. For each $\theta>0$, define the probability measure $P_\theta$ on $A$ by $$\frac{dP_\theta}{dQ}(x) = \frac{\exp{(-H(x)/\theta)}}{\int \exp{(-H(x)/\theta)}dQ(x)}.$$ Then under certain regularity conditions of $H$, Chii-Ruey Hwang's paper shows that the measures $(P_\theta)_{\theta>0}$ converges weakly to a probability measure supported on the minimal set $$N = \{x\in A | H(x) = \min_{y\in A} H(y) \}. $$
Now suppose that I have a sequence of continuous and bounded functions $(H_\theta)_{\theta>0}$ converges uniformly to $H$ on $A$. I was wondering whether the same weak convergence result still holds for the following Gibbs measures: $$\frac{dP_\theta}{dQ}(x) = \frac{\exp{(-H_\theta(x)/\theta)}}{\int \exp{(-H_\theta(x)/\theta)}dQ(x)}.$$ Has such a problem been studied in the literature (say the large derivation theory)? If so, could you provide me a reference for it?
