  We reconcile for the first time the strict mathematical formalism of
multivariate cumulants with the usage of cumulants in anisotropic flow analyses
in high-energy nuclear collisions. This reconciliation yields to the next
generation of estimators to be used in flow analyses. We review all fundamental
properties of multivariate cumulants and use them as a foundation to establish
two simple necessary conditions to determine whether some multivariate random
variable is a multivariate cumulant in the basis they are expressed in. We
argue that properties of cumulants are preserved only for the stochastic
variables on which the cumulant expansion has been performed directly, and if
there are no underlying symmetries due to which some terms in the cumulant
expansion are identically zero. We illustrate one possibility of new
multivariate cumulants of azimuthal angles by defining them event-by-event and
by keeping all non-isotropic terms in the cumulant expansion. Further, we
introduce new cumulants of flow amplitudes named Asymmetric Cumulants, which
generalize recently introduced Symmetric Cumulants for the case when flow
amplitudes are raised to different powers. Finally, we present the new concept
of Cumulants of Symmetry Plane Correlations and provide the first realisation
for the lowest orders. The new estimators can be used directly to constrain the
multivariate probability density function of flow fluctuations, since its
functional form can be reconstructed only from its true moments or cumulants.
The new definition for cumulants of azimuthal angles enables separation of
nonflow and flow contributions, and offers first insights into how the
combinatorial background contributes for small multiplicities to flow
measurements with correlation techniques. All the presented results are
supported by Monte Carlo studies using state-of-the-art models.
