I have a statistic that can be written as the average of ergodic random variables identically distributed. This statistic depends on a parameter. It is known that the statistic converges in probability to the common expectation of the random variables. I am wondering if the convergence is uniform.
Here is the problem.
Let $X_1, \dots, X_n$ be a sequence of ergodic real-valued random variables identically distributed. Let $\theta \in \Theta$, where $\Theta$ is a compact subset of $\mathbb{R}$. I consider the statistic $S_n(\theta)$ defined by
$$S_n(\theta) = \dfrac{1}{n}\sum_{i = 1}^n f(X_i, \theta),$$ where $f$ is a bounded function on $\mathbb{R}^2$. Can I say that $S_n(\theta)$ uniformly converges to $S_0(\theta) = E(S_n(\theta))$?
I read somewhere that sufficient conditions for the uniform convergent are: $\Theta$ compact and $f$ bounded. But I am not very sure.