  Fluctuation measurements are important sources of information on the
mechanism of particle production at LHC energies. This article reports the
first experimental results on third-order cumulants of the net-proton
distributions in Pb$-$Pb collisions at a center-of-mass energy $\sqrt{s_{\rm
NN}} = 5.02$ TeV recorded by the ALICE detector. The results on the
second-order cumulants of net-proton distributions at $\sqrt{s_{\rm NN}} =
2.76$ and $5.02$ TeV are also discussed in view of effects due to the global
and local baryon number conservation. The results demonstrate the presence of
long-range rapidity correlations between protons and antiprotons. Such
correlations originate from the early phase of the collision. The experimental
results are compared with HIJING and EPOS model calculations, and the
dependence of the fluctuation measurements on the phase-space coverage is
examined in the context of lattice quantum chromodynamics (LQCD) and hadron
resonance gas (HRG) model estimations. The measured third-order cumulants are
consistent with zero within experimental uncertainties of about 4% and are
described well by LQCD and HRG predictions.
