  We connect shift-invariant characteristic kernels to infinitely divisible
distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important
role in machine learning applications with their kernel means to distinguish
any two probability measures. The contribution of this paper is two-fold.
First, we show, using the L\'evy-Khintchine formula, that any shift-invariant
kernel given by a bounded, continuous and symmetric probability density
function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is
characteristic. We also present some closure property of such characteristic
kernels under addition, pointwise product, and convolution. Second, in
developing various kernel mean algorithms, it is fundamental to compute the
following values: (i) kernel mean values $m_P(x)$, $x \in \mathcal{X}$, and
(ii) kernel mean RKHS inner products ${\left\langle m_P, m_Q
\right\rangle_{\mathcal{H}}}$, for probability measures $P, Q$. If $P, Q$, and
kernel $k$ are Gaussians, then computation (i) and (ii) results in Gaussian
pdfs that is tractable. We generalize this Gaussian combination to more general
cases in the class of infinitely divisible distributions. We then introduce a
{\it conjugate} kernel and {\it convolution trick}, so that the above (i) and
(ii) have the same pdf form, expecting tractable computation at least in some
cases. As specific instances, we explore $\alpha$-stable distributions and a
rich class of generalized hyperbolic distributions, where the Laplace, Cauchy
and Student-t distributions are included.
