I am new to convolution. Below is some derivation related to convolution I saw in a paper. Hope to get some help here. (The paper is "Comparing nonparametric and parametric regresssion fit" published in "The Annals of Statistics"(1993).)
There are two specific questions I want to ask. Here are some notations for both questions.
$K_{h}(\cdot) = h^{-1} K(\cdot/h)$, where $h\equiv h_{n}$ is a bandwidth (decreasing to 0 at the rate $n^{-1/5}$), and $K$ is a "kernel" function satisfying: $K$ is symmetric, twice continuously differentiable with compact support, and $\int K(u)du = 1$. Let $K^{(j)}_{h}$ denote the $j-$times convolution product of $K_{h}$.
My first QUESTION is to verify the following equation given in the paper.
$$\int K_{h}(u_{1}-u_{2}) K_{h}(u_{2}-u_{3}) K_{h}(u_{3}-u_{4}) K_{h}(u_{4}-u_{1})du_{1}\ldots du_{4} = \int (K^{(2)}_{h}(u))^{2}du. \quad \clubsuit$$
Here is what I derived for $\clubsuit$. Since $K(\cdot)$ is symmetric, \begin{align} &\int K_{h}(u_{1}-u_{2}) K_{h}(u_{2}-u_{3}) K_{h}(u_{3}-u_{4}) K_{h}(u_{4}-u_{1})du_{1}\ldots du_{4} \\ = & \int K_{h}(s) K_{h}(u-s) K_{h}(s^{\prime}) K_{h}(u-s^{\prime})\\ = & \int (K^{(2)}_{h}(u))^{2}du \end{align} I am not sure if this is correct or if there is a better way to show this.
My second QUESTION is to verify the following relation also given in the paper.
$$O(\frac{h^{2}}{n^{4}}) \int K_{h}(u)^{2}K^{(2)}_{h}(u)du = O(\frac{1}{n^{4}}) = o(\frac{1}{n^{3}}). \quad \spadesuit$$ (Here $O$ and $o$ are the typical notations for comparing magnitude orders.)
If it were written without $K^{(2)}_{h}(u)$ in the integral, I could easily understand it, as $\int K_{h}(u)^{2}du=h^{-2} \int K(u)^{2}du = O(h^{-2})$. This is why I feel doubtful about $\spadesuit$ given in the paper.