  We introduce a general procedure to pave the road towards the ultimate goal
of deriving analytic expressions for the probability density functions
(p.d.f.'s) of multiparticle azimuthal correlations. All multiparticle azimuthal
correlators can be expressed analytically in terms of the real and imaginary
parts of $M$-particle $Q$-vectors. We derive the analytic results for the
p.d.f.'s of single-particle $Q$-vectors in the most general case and
demonstrate that they can be expressed solely in terms of Chebyshev polynomials
of the first kind. This leads analytically to the expressions of the
characteristic functions of $M$-particle $Q$-vectors in terms of Bessel
functions of the first kind. From the obtained characteristics functions we
calculate the higher order moments of the real and imaginary parts of
$M$-particle $Q$-vectors and use them to obtain the higher order moments of
multiparticle azimuthal correlators. Finally, these results are used to
investigate the sensitivity of multiparticle azimuthal correlations and to
illuminate requirements necessary for future anisotropic flow measurements.
