  At high angular frequencies, beyond the damping tail of the primary power
spectrum, the dominant contribution to the power spectrum of cosmic microwave
background (CMB) temperature fluctuations is the thermal Sunyaev-Zel'dovich
(tSZ) effect. We investigate various important statistical properties of the
Sunyaev-Zel'dovich maps, using well-motivated models for dark matter clustering
to construct statistical descriptions of the tSZ effect to all orders enabling
us to determine the entire probability distribution function (PDF). Any generic
deterministic biasing scheme can be incorporated in our analysis and the
effects of projection, biasing and the underlying density distribution can be
analysed separately and transparently in this approach. We introduce the
cumulant correlators as tools to analyse tSZ catalogs and relate them to
corresponding statistical descriptors of the underlying density distribution.
The statistics of hot spots in frequency-cleaned tSZ maps are also developed in
a self-consistent way to an arbitrary order, to obtain results complementary to
those found using the halo model. We also consider different beam sizes, to
check the extent to which the PDF can be extracted from various observational
configurations. The formalism is presented with two specific models for
underlying matter clustering: (1) the hierarchical ansatz; and (2) the
lognormal distribution. We find both models to be in very good agreement with
the simulation results, though the lognormal model has an edge over the
hierarchical model.
