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This article studies large and local large deviations for sums of i.i.d.real-valued random variables in the domain of attraction of an $\alpha$-stablelaw, $\alpha\in (0,2]$, with emphasis on the case $\alpha=2$. There are twodifferent scenarios: either the deviation is realised via a collectivebehaviour with all summands contributing to the deviation (a Gaussianscenario), or a single summand is atypically large and contributes to thedeviation (a one-big-jump scenario). Such results are known when $\alpha \in(0,2)$ (large deviations always follow a one big-jump scenario) or when therandom variables admit a moment of order $2+\delta$ for some $\delta>0$. Weextend these results, including in particular the case where the right tail isregularly varying with index $-2$ (treating cases with infinite variance in thedomain of attraction of the normal law). We identify the threshold for thetransition between the Gaussian and the one-big-jump regimes; it is slightlylarger when considering local large deviations compared to integral largedeviations. Additionally, we complement our results by describing the behaviourof the sum and of the largest summand conditionally on a (local) largedeviation, for any $\alpha\in (0,2]$, both in the Gaussian and in theone-big-jump regimes. As an application, we show how our results can be used inthe study of condensation phenomenon in the zero-range process at the criticaldensity, extending the range of parameters previously considered in theliterature.

Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon