Consider $c_n, h>0$, $K:\mathbb{R}^d\to\mathbb{R}$ and $f>0$ the density function of the random variable $X_0$. The norm $\lVert u\rVert=\max(\lvert u_1\rvert,\dotsc,\lvert u_d\rvert)$ is the maximum norm. I want to understand why this inequality holds: (the integrand is just a positive function of $u$)
The author says "using region of integration". I see this as the integral of a function of $u$ on the region outside the hypercube with side $c_n$. Why is this smaller than the integral on the whole $\mathbb{R}^d$ but with the function multiplied by $(\lVert u\rVert/c_n)^q$?
Thanks in advance!
Considerations
For simplicity, let $g\geq 0$ be the integrand. Then, $$\int_{\lVert u\rVert>c_n}g(u)du\leq \int_{\lVert u\rVert>c_n}(\lVert u\rVert/c_n)^d g(u)du\leq \int_{\mathbb{R}^d}(\lVert u\rVert/c_n)^d g(u)du$$