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Ultra-high energy cosmic rays (UHECRs) interacting with the atmospheregenerate extensive air showers (EAS) of secondary particles. The depthcorresponding to the maximum development of the shower, $\Xmax$, is awell-known observable for determining the nature of the primary cosmic raywhich initiated the cascade process. In this paper, we present an empiricalmodel to describe the distribution of $\Xmax$ for EAS initiated by nuclei, inthe energy range from $10^{17}$ eV up to $10^{21}$ eV, and by photons, in theenergy range from $10^{17}$ eV up to $10^{19.6}$ eV. Our model adopts thegeneralized Gumbel distribution motivated by the relationship between thegeneralized Gumbel statistics and the distribution of the sum ofnon-identically distributed variables in dissipative stochastic systems. Weprovide an analytical expression for describing the $\Xmax$ distribution forphotons and for nuclei, and for their first two statistical moments, namely$\langle \Xmax\rangle$ and $\sigma^{2}(\Xmax)$. The impact of the hadronicinteraction model is investigated in detail, even in the case of the mostup-to-date models accounting for LHC observations. We also briefly discuss thedifferences with a more classical approach and an application to theexperimental data based on information theory.

Reinterpreting the development of extensive air showers initiated by nuclei and photons