Order of a kernel

In statistics, the order of a kernel is the degree of the first non-zero moment of a kernel.^[1]

The literature knows two major definitions of the order of a kernel. Namely are:

Let ${\displaystyle \ell \geq 1}$ be an integer. Then, ${\displaystyle K:\mathbb {R} \rightarrow \mathbb {R} }$ is a kernel of order ${\displaystyle \ell }$ if the functions ${\displaystyle u\mapsto u^{j}K(u),~j=0,1,...,\ell }$ are integrable and satisfy ${\displaystyle \int K(u)du=1,~\int u^{j}K(u)du=0,~~j=1,...,\ell .}$^[2]