### Sums of i.i.d. random variables: a phase transition between Gaussian and one-big-jump regimes

Probability Seminar

10th May 2024, 3:30 pm – 4:30 pm

Fry Building, 2.04

For large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an alpha-stable law with alpha in (0,2], there are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when alpha in (0,2) or when alpha=2 and the random variables admit a moment of order 2+delta for some delta>0 with the right-tail probability regularly varying.

This talk will present new results extending the above known phenomenon to the last missing case including where alpha=2 and the right tail is regularly varying with index -2. Threshold for the phase transition between the Gaussian and the one-big-jump regimes will be identified and applications will be discussed.

This talk is based on a joint work with Quentin Berger (Paris) and Matthias Birkner (Mainz).

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