  The use of correntropy as a similarity measure has been increasing in
different scenarios due to the well-known ability to extract high-order
statistic information from data. Recently, a new similarity measure between
complex random variables was defined and called complex correntropy. Based on a
Gaussian kernel, it extends the benefits of correntropy to complex-valued data.
However, its properties have not yet been formalized. This paper studies the
properties of this new similarity measure and extends this definition to
positive-definite kernels. Complex correntropy is applied to a channel
equalization problem as good results are achieved when compared with other
algorithms such as the complex least mean square (CLMS), complex recursive
least squares (CRLS), and least absolute deviation (LAD).
