Order of a kernel

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

Definitions

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

Definition 1

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