On the convergence of the infimum of an average

Question

The strong law of large numbers states that the sample average converges almost surely to the expected value: $\frac{1}{n}\sum_{i=1}^ng(X_i,t)\rightarrow\mathbf{E}[g(X,t)]$ (a.s) for all $t\in\mathbb{R}$. I want to estimate: $$C=\inf_{t>0}\frac{1}{t}\log\left(\mathbf{E}[g(X,t)]\right)$$

I propose $\hat{C}_n=\inf_{t>0}\frac{1}{t}\log\left(\frac{1}{n}\sum_{i=1}^ng(X_i,t)\right)$. Is $\hat{C}_n\rightarrow C$ convergence true? Do I have to use any additional assumptions to ensure it (besides that the expectation exists)?

Answer 1

You need continuity, $t$ must lie in a non-trivial compact set, and uniform convergence. See the Chapter by Newey and McFadden in the Handbook of Econometrics for intuition. There are many, many statistical proofs around.