  The number density of local maxima of weak lensing field, referred to as
weak-lensing peak counts, can be used as a cosmological probe. However, its
relevant cosmological information is still unclear. We study the relationship
between the peak counts and other statistics in weak lensing field by using
1000 ray-tracing simulations. We construct a local transformation of lensing
field $\cal K$ to a new Gaussian field $y$, named local-Gaussianized
transformation. We calibrate the transformation with numerical simulations so
that the one-point distribution and the power spectrum of $\cal K$ can be
reproduced from a single Gaussian field $y$ and monotonic relation between $y$
and $\cal K$. Therefore, the correct information of two-point clustering and
any order of moments in weak lensing field should be preserved under
local-Gaussianized transformation. We then examine if local-Gaussianized
transformation can predict weak-lensing peak counts in simulations. The
local-Gaussianized transformation is insufficient to explain weak-lensing peak
counts in the absence of shape noise. The prediction by local-Gaussianized
transformation underestimates the simulated peak counts with a level of
$\sim20-30\%$ over a wide range of peak heights. Local-Gaussianized
transformation can predict the weak-lensing peak counts with a $\sim10\%$
accuracy in the presence of shape noise. Our analyses suggest that the
cosmological information beyond power spectrum and its moments would be
necessary to predict the weak-lensing peak counts with a percent-level
accuracy, which is an expected statistical uncertainty in upcoming wide-field
galaxy surveys.
