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Higher order moments of multiparticle azimuthal correlations

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.