  In this paper, we develop a method of performing the one-point statistics of
a perturbed density field with a multiresolutional decomposition based on the
discrete wavelet transform (DWT). We establish the algorithm of the one-point
variable and its moments in considering the effects of Poisson sampling and
selection function. We also establish the mapping between the DWT one-point
statistics in redshift space and real space, i.e. the algorithm for recovering
the DWT one-point statistics from the redshift distortion of bulk velocity,
velocity dispersion, and selection function. Numerical tests on N-body
simulation samples show that this algorithm works well on scales from a few
hundreds to a few Mpc/h for four popular cold dark matter models.
  Taking the advantage that the DWT one-point variable is dependent on both the
scale and the shape (configuration) of decomposition modes, one can design
estimators of the redshift distortion parameter (beta) from combinations of DWT
modes. When the non-linear redshift distortion is not negligible, the beta
estimator from quadrupole-to-monopole ratio is a function of scale. This
estimator would not work without adding information about the scale-dependence,
such as the power-spectrum index or the real-space correlation function of the
random field. The DWT beta estimators, however, do not need such extra
information. Numerical tests show that the proposed DWT estimators are able to
determine beta robustly with less than 15% uncertainty in the redshift range 0
< z < 3.
