Large Deviation Principles for Sequences of Maxima and Minima

Large deviations of the maximum of independent and identically distributed random variables

“…It is also easy to deduce along the same lines that the rate function ψ r (ξ) for the maximum of i.i.d. variables whose common pdf decays at infinity as p(x) ∼ e −x δ is given by ψ r (ξ) = ξ δ − 1, in agreement with [10]. I am not aware of a similar LDT treatment for EVS of densities in the Fréchet basin of attraction, while for the Weibull class this is also possible (with speed N ), but somewhat much less interesting [10].…”

“…Much to my surprise, I was able to retrieve only a single paper [10] where the EVS of i.i.d. random variables was looked at through the prism of LDT.…”

The joint distribution of the maximum and minimum of an AR(1) process

“…However, the recent years have seen considerable developments on the asymptotics of the joint distribution of M n and m n following on from Davis (1979). We mention: the joint distribution of M n and m n for complete and incomplete samples of stationary sequences (Peng et al, 2010); the joint distribution of M n and m n for complete and incomplete samples from weakly dependent stationary sequences (Peng et al, 2011); the joint distribution of M n and m n for strongly dependent Gaussian vector sequences (Weng et al, 2012); the joint distribution of M n and m n for independent spherical processes (Hashorva, 2013); the joint distribution of M n and m n for dependent stationary Gaussian arrays (Hashorva and Weng, 2013); large deviation results on M n and m n for independent and identical samples (Giuliano and Macci, 2014); the joint distribution of M n and m n for scaled stationary Gaussian sequences (Hashorva et al, 2014); the joint distribution of M n and m n for complete and incomplete stationary sequences (Hashorva and Weng, 2014a,b); the joint distribution of M n and m n for Hüsler-Reiss bivariate Gaussian arrays (Liao and Peng, 2014). None of these papers give the exact joint distribution of M n and m n for finite n.…”

Real-space renormalization for disordered systems at the level of large deviations

“…The typical fluctuations are classified in terms of the three universality classes Gumbel-Fréchet-Weibull [36,37], with many applications in various physics domains (see the reviews [38][39][40] and references therein) and have been much studied from the renormalization perspective [41][42][43][44][45]. The large deviations properties of the empirical maximum have been found to be asymmetric [35,46,47], as a consequence of the following obvious asymmetry : an 'anomalously good' maximum requires only one anomalously good variable, while an 'anomalously bad' maximum requires that all variables are anomalously bad. This simple argument allows to understand why the large deviations will be also completely different for the two tails x → ±∞ in the more general problem of Eq.…”