I am reading materials about nonparametric estimation in statistics. I have some concerns about the conditions when making approximation to one integral.
Suppose $h_n \rightarrow 0$ as $n \rightarrow \infty$ and $f$ has bounded $m$th order derivative. Also $\int K(u)\,du=1$ and $\int u^iK(u) \,du =0$ for $i=1,2,\ldots,m-1$. We want to explore the property of the integral below as $n$ is large enough. The $\int$ is from $-\infty$ to $+\infty$. Below is the regular way to deal with it:
For a given $x$:
\begin{align} &\int \frac{1}{h_n} K\left(\frac{x-y}{h_n}\right)f(y)\,dy\\ ={}&\int K(u)f(x+h_nu)\,du \\ ={}&\int K(u)\left[f(x)+\sum_{i=1}^m f^{(i)}(x)(h_nu)^i + o({(h_nu)}^m)\right]\,du\\ ={}&f(x)+f^{(m)}(x)h_n^m\int u^mK(u)\,du + o(h_n^m) \end{align}
I feel uncomfortable on the little-o thing in the last equation. I think it's a problem of interchanging limitation and integrals. I was wondering do we need some conditions to make the last equation hold? Thanks!
This isn't a complete answer, but there are a few things you can try out:
$(1)$ Try making use of Lebesgue's Dominated Convergence Theorem and try to find some limit $h_n\rightarrow h$ and relate it to the integral in question.
$(2)$ Since f has bounded $m$-order derivatives, you may try to bound your integral. If the derivatives are bounded, then the functions are all Lipschitz continuous, so you get an upper bound. Again, you may want to consider $h_n$ where all the functions have some "nice properties" (i.e. $C^{k}$ differentiability, smoothness, compact support etc.).
$(3)$ pull the $\frac{1}{h_n}$ outside the integral and then compare this sequence to the functions that are being integrated. For the inside argument, a convolution argument might work.