The statistical property of the weak lensing fields is studied quantitatively using the ray-tracing simulations .
Motivated by the empirical lognormal model that characterizes the probability distribution function of the three-dimensional mass distribution excellently , we critically investigate the validity of lognormal model in the weak lensing statistics .
Assuming that the convergence field , \kappa , is approximately described by the lognormal distribution , we present analytic formulae of convergence for the one-point probability distribution function ( PDF ) and the Minkowski functionals .
The validity of lognormal models is checked in detail by comparing those predictions with ray-tracing simulations in various cold dark matter models .
We find that the one-point lognormal PDF can describe the non-Gaussian tails of convergence fields accurately up to \nu \sim 10 , where \nu is the level threshold given by \nu \equiv \kappa / \langle \kappa ^ { 2 } \rangle ^ { 1 / 2 } , although the systematic deviation from lognormal prediction becomes manifest at higher source redshift and larger smoothing scales .
The lognormal formulae for Minkowski functionals also fit to the simulation results when the source redshift is low , z _ { s } = 1 .
Accuracy of the lognormal-fit remains good even at the small angular scales 2 ’ \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } \theta \lower 2.15 pt \hbox { $ % \buildrel < \over { \sim } $ } 4 ’ , where the perturbation formulae by Edgeworth expansion break down .
On the other hand , the lognormal model enables us to predict the higher-order moments , i.e. , skewness , S _ { 3 , \kappa } and kurtosis , S _ { 4 , \kappa } , and we thus discuss the consistency by comparing the predictions with the simulation results .
Since these statistics are very sensitive to the high ( low ) -convergence tails , the lognormal prediction does not provide a quantitative successful fit .
We therefore conclude that the empirical lognormal model of the convergence field is safely applicable as a useful cosmological tool , as long as we are concerned with the non-Gaussianity of \nu \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 5 for low- z _ { s } samples .