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We studied the gauge invariant cumulants associated with the Zak phase. We have shown that for localized Wannier functions they correspond to the cumulants of the Wannier centers. They are also related to the dielectric response functions of a given system. We calculated the cumulants for the Rice-Mele model. In the limit of isolated dimers, all the moments (extracted from the gauge invariant cumulants) are equal. This can be justified for this case by constructing the operator which corresponds to the Berry phase explicitly. Deviations from this behavior come about when the hopping parameters are both finite. For a system with equal hopping parameters the odd cumulants vanish. We have also reconstructed the full probability distribution of the polarization based on the gauge invariant cumulants and have studied how they evolve as functions of different parameters of the Hamiltonian. In particular we calculated the evolution of the distribution around the topologically non-trivial point of the model. We anticipate that detailed experimental measurements can also provide a probability distribution of the polarization for comparison with our predictions.

Acknowledgments {#acknowledgments .unnumbered}
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The authors acknowledge financial support from the Turkish agency for basic research (TÜBITAK, grant no. 113F334). We also thank 