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Event-by-event elliptic flow harmonic distributions $p(v_{2})$ are measured in PbPb collisions at $\sqrt{s_{NN}} = 5.02$~TeV using the CMS detector for the integrated $p_{T}$ range $0.3 < p_{T} < 3.0$~GeV/$c$ and pseudorapidity range $\left| eta \right|<1.0$. In order to gain insight on the nature of the initial geometry fluctuations, cumulant flow harmonics are calculated from the moments of the $v_2$ probability distribution $p(v_{2})$. A fine-level splitting between the higher-order cumulants is observed. The skewness with respect to the reaction plane is estimated from the cumulants and found to be negative, which suggests a non-Gaussian nature for the initial-state fluctuations, as predicted by hydrodynamic models. These observations suggest a non-Gaussian nature of the initial-state fluctuations. Furthermore, assuming that the flow harmonics are linearly proportional to the initial-state eccentricities, the $p(v_{2})$ distributions are fitted using an elliptic power law parameterization to study the initial-state geometry fluctuations.