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Mikosch, Thomas; Nagaev, A. V.

doi:10.1023/a:1009913901219

Cited by 150 publications

(5 citation statements)

References 32 publications

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“…As a consequence of diverging second (or even first) moments of the distribution of the stochastic variable of reference, the probability of sums decays sub-exponentially and the rate function is trivially 0 (Touchette 2009). Large deviation results can still be obtained in many cases; however, they are dominated by the largest values in the sample instead of the mean, as already mentioned in the introduction (Mikosch and Nagaev 1998). These conditions are relevant for some variables of interest in (geo)physical fluid dynamics.…”

Section: Constructing the Rate Functions Describing The Large Deviationsmentioning

confidence: 90%

“…This will roughly be, in fact, the scenario we will explore below. However, the link between persistence and extremes of finite-size averages is not always true: in the case of heavy-tailed random variables, for example, the extremes of averages are dominated by a single very large extreme event within the averaging window (Mikosch and Nagaev 1998). We remark that, generally, the methods of extreme value theory can also be applied in the same way to study extremes of averaged observables.…”

Section: A Mathematical Framework For Climatic Extremesmentioning

confidence: 97%

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A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere

2019

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We study temporally persistent and spatially extended extreme events of temperature anomalies, i.e. heat waves and cold spells, using large deviation theory. To this end, we consider a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature in the mid-latitudes. We find that, after a re-normalisation based on the integrated auto-correlation, the rate function one obtains at a given latitude by looking locally in space at long time averages agrees with what is obtained, instead, by looking locally in time at large spatial averages along the latitude. This is a result of scale symmetry in the spatio-temporal turbulence and of the fact that advection is primarily zonal. This agreement hints at the universality of large deviations of the temperature PAPER: Interdisciplinary statistical mechanics

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Section: Constructing the Rate Functions Describing The Large Deviationsmentioning

confidence: 90%

Section: A Mathematical Framework For Climatic Extremesmentioning

confidence: 97%

A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere

2019

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“…To show that these probabilities are of order we only need that (since the remaining probability in Lemma 3 can be estimated with Hoeffding’s inequality, see Hoeffding ( 1963 )) with in Lemma 4 and in Lemma 3 . The asymptotics (3.2) in Mikosch and Nagaev ( 1998 ) states that . Consequently, we need to choose such that as well as .…”

Section: Resultsmentioning

confidence: 97%

Haldane’s formula in Cannings models: the case of moderately strong selection

et al. 2021

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For a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

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“…In this case, the description of the large fluctuations requires more precise asymptotic results for specific classes of waiting times. Still regarding renewal-reward processes as primary tools, we mention that precise large deviation principles for renewal-reward processes have been established when rewards are independent of waiting times for several types of heavy-tailed waiting times and rewards [52][53][54][55][56][57]. Some forms of dependence between rewards and waiting times have been considered in [59,60].…”

Section: Positive Occupation Time Of the Reset Brownian Motionmentioning

confidence: 99%

Statistical fluctuations under resetting: rigorous results

Zamparo¹

2022

J. Phys. A: Math. Theor.

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In this paper we investigate the normal and the large fluctuations of additive functionals associated with a stochastic process under a general non-Poissonian resetting mechanism. Cumulative functionals of regenerative processes are very close to renewal-reward processes and inherit most of the properties of the latter. Here we review and use the classical law of large numbers and central limit theorem for renewal-reward processes to obtain same theorems for additive functionals of a stochastic process under resetting. Then, we establish large deviation principles for these functionals by illustrating and applying a large deviation theory for renewal-reward processes that has been recently developed by the author. We discuss applications of the general results to the positive occupation time, the area, and the absolute area of the reset Brownian motion. While introducing advanced tools from renewal theory, we demonstrate that a rich phenomenology accounting for dynamical phase transitions emerges when one goes beyond Poissonian resetting.

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Cited by 150 publications

References 32 publications

A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere

A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere

Haldane’s formula in Cannings models: the case of moderately strong selection

Statistical fluctuations under resetting: rigorous results

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