Let $X(\lambda_n^j)$ be a random variable defined on $[0, \infty)$ and $\lambda_n^j$ be the Fourier frequencies $\frac{2 \pi j }{n}$, $j = 0,...n$. I will denote $X(\lambda_{j,n}) =: X_n^j$ and I know that $X_n^j$ converges weakly to $X_j$ (for each $j$) and $X_0, X_1, ....$ are i.i.d. rv.
I need to prove the following uniform convergence
$$\sup _\theta | \frac{1}{n} \sum_j^n w( \lambda_n^j, \theta) f( \theta, \lambda_n^j, X_n^j ) - \frac{1}{2 \pi} \int_0^{2\pi} w(\lambda, \theta) E(f(\theta, \lambda, X_0)) d \lambda$$
What I know so far:
- $\theta \in \Theta$ which is a compact set.
- the weights $w(\lambda_n^j, \theta))$ are differentiable in both component with bounded derivatives.
Hence, I guess that an equicontinuity argument (or the mean value theorem?) and the properties of the Riemann integral will allow me to state that
$$(1) \sup _\theta |\frac{1}{n} \sum_j^n w(\lambda_n^j, \theta) - \frac{1}{2 \pi}\int_0^{2\pi} w(\lambda; \theta) d \lambda| \rightarrow 0$$
even though I am not quite sure how to prove it.
I managed also to prove that for some functions $f$ that satisfy some conditions
$$(2) \sup _\theta |Var(f(\theta, \lambda_n^j, X_n^j)) - Var(f(\theta, \lambda_n^j, X_0))| \rightarrow 0$$
My questions are:
- How can I exactly formulate the argument for (1)?
- How could I combine (1) and (2) together to prove the initial uniform convergence.