Integral Limit Theorems Taking Large Deviations Into Account When Cramér’s Condition Does Not Hold. II

Nagaev, A. V.

doi:10.1137/1114028

Cited by 100 publications

(114 citation statements)

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“…In Section 3 we prove large deviation results for Minkowski sums S n of iid regularly varying random compact sets. To the best of our knowledge, such results are not available in the literature; they parallel those proved by A. and S. Nagaev [10,11] for sums of iid random variables. The case of general random compact sets is treated in [8].…”

Section: Lemma 1 (I)supporting

confidence: 77%

“…for large n. Therefore, (10) says that I 21 = 0 for large n. Furthermore, we have already established in the proof of Theorem 1 that γ n I 22 −→0 as n → ∞ if δ is small enough, relative to τ . The statement of the proposition follows.…”

Section: Remarkmentioning

confidence: 66%

“…Large deviations results for sums of heavy-tailed random elements significantly differ from Cramér-type results. In this case it is typical that only the largest summand determines the large deviation behavior; see the classical results by A. Nagaev [10] for sums of iid random variables; cf. [11,6].…”

Section: Introductionmentioning

confidence: 93%

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A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Mikosch¹,

Pawlas²,

Samorodnitsky³

2011

J. Appl. Probab.

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We prove large deviation results for Minkowski sums Sn of iid random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: "large" values of the sum are essentially due to the "largest" summand. These results extend those in [8] for generally non-convex sets, where we assumed that the normalization of Sn grows faster than n.

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Section: Lemma 1 (I)supporting

confidence: 77%

Section: Remarkmentioning

confidence: 66%

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A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Mikosch¹,

Pawlas²,

Samorodnitsky³

2011

J. Appl. Probab.

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“…In this context, Nagaev [16] has proved that (8) is satisfied as long as N = ρ c L + γ (L)L 1 2λ with γ (L) → ∞ as L → ∞. In view of equations (11) and (12), we may choose the sequence C L = √ L log L in the line following the expression (9), and adapt the arguments presented in the previous section to prove the following theorem.…”

Section: Remarksmentioning

confidence: 95%

Thermodynamic limit for the invariant measures in supercritical zero range processes

Armendáriz

Loulakis

2008

Probab. Theory Relat. Fields

112

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We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.

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“…Ce résultat, qui traite des écarts modérés (i.e. du comportement de S n dans un domaine de taille √ cn log n), a été démontré par Rubin et Sethuraman dans [RuS] (voir aussi [Nag2,Nag3], [Mic] et [Sla]). Cependant comme nous en aurons besoin dans la suite de cet article et que l'argument que nous présentons est très différent de l'argument original et se généralise automatiquement en dimension supérieure, nous choisissons de l'écrire ici.…”

Section: Loi Des éCarts Modérésunclassified

Distributions diophantiennes et théorème limite local sur

Breuillard

2004

Probab. Theory Relat. Fields

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We study the speed of convergence of n d/2 f dµ * n in the local limit theorem on R d under very general conditions upon the function f and the distribution µ. We show that this speed is at least of order 1 n and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f . We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks.

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Integral Limit Theorems Taking Large Deviations Into Account When Cramér’s Condition Does Not Hold. II

Cited by 100 publications

References 1 publication

A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Thermodynamic limit for the invariant measures in supercritical zero range processes

Distributions diophantiennes et théorème limite local sur

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