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Pierre Petit

Publications and source records attributed to Pierre Petit.

9 recordsLinked to original sources
arXiv2021

Probabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate function at the transition

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when x n > n 1/2 (see, [13, 14]). In this paper, we derive rough asymptotics results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition.

math.PR↗
arXiv2020

Large deviation results for triangular arrays of semiexponential random variables

Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when X 1 has a semiexponential distribution (see, [16, 17]). In the same setting, the authors of [4] derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition. In this paper, we exhibit the same asymptotic behaviour for triangular arrays of semiexponentially distributed random variables, no more supposed absolutely continuous.

math.PR↗
arXiv2015

A conditional Berry-Esseen bound and a conditional large deviation result without Laplace transform. Application to hashing with linear probing

\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.

math.PR↗
arXiv2011

Cramér's theorem for asymptotically decoupled fields

We give a general setting for Cramér's large deviations theorem for the empirical means of a field of random vectors, which contains Cramér's theorem for i.i.d. random vectors and Sanov's theorem for asymptotically decoupled measures. ----- Nous établissons un cadre général pour le théorème de Cramér sur les grandes déviations des moyennes empiriques d'un champ de vecteurs aléatoires, cadre qui contient le théorème de Cramér pour des vecteurs aléatoires i.i.d. et le théorème de Sanov pour les mesures asymptotiquement découplées.

math.PR↗
arXiv2011

Cramér's theorem in measurable locally convex spaces

We give a general setting for Cramér's large deviations theorem for the empirical means of a sequence of i.i.d. random vectors, which contains Cramér's theorem in a Banach space and Sanov's theorem. ----- Nous établissons un cadre général pour le théorème de Cramér sur les grandes déviations des moyennes empiriques d'une suite de vecteurs aléatoires i.i.d., cadre qui contient le théorème de Cramér dans les espaces de Banach séparables et le théorème de Sanov.

math.PR↗
arXiv2011

A linear version of Dawson-Gärtner's theorem

We prove a linear version of Dawson-Gärtner's theorem saying that weak large deviations principles and the equivalence of ensembles are preserved through linear projective limits. ----- Nous démontrons une version linéaire du théorème de Dawson-Gärtner assurant que les principes de grandes déviations faibles et l'équivalence d'ensemble sont conservés par passage aux limites projectives d'espaces vectoriels.

math.PR↗
arXiv2011

A proof of the equivalence of ensembles for asymptotically decoupled fields relying on Mosco's theorem

We give a proof of the identification between the entropy and the opposite of the Fenchel-Legendre transform of the pressure for asymptotically decoupled fields relying on Mosco's theorem. ----- Nous démontrons l'identification entre l'entropie et l'opposée de la transformée de Fenchel-Legendre de la pression pour les champs asymptotiquement découplés à partir du théorème de Mosco.

math.PR↗
arXiv2004

On the diffraction pattern of C60 peapods

We present detailed calculations of the diffraction pattern of a powder of bundles of C$_{60}$ peapods. The influence of all pertinent structural parameters of the bundles on the diffraction diagram is discussed, which should lead to a better interpretation of X-ray and neutron diffraction diagrams. We illustrate our formalism for X-ray scattering experiments performed on peapod samples synthesized from 2 different technics, which present different structural parameters. We propose and test different criteria to solve the difficult problem of the filling rate determination.

cond-mat.other↗