On Large Deviations from the Invariant Measure

Gärtner, Jürgen

doi:10.1137/1122003

Cited by 280 publications

(227 citation statements)

References 8 publications

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“…By the Girtner (71-Ellis (61 theorem, under conditions (2) and (4)- (7). the large deviations principle (LDP) bolds for Z with large deviations rate function I; i.e., for each Borel set A …”

Section: Let I Be the Associated Large Deviations (Lld) Rate Functionmentioning

confidence: 99%

Large deviations behavior of counting processes and their inverses

Glynn

Whitt

1994

Queueing Syst

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Section: Let I Be the Associated Large Deviations (Lld) Rate Functionmentioning

confidence: 99%

Large deviations behavior of counting processes and their inverses

Glynn

Whitt

1994

Queueing Syst

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“…2. In the theory and applications of large deviations, I plays an important role as the rate function for the normalised occupation measure of one Brownian motion (or, one Brownian bridge) in , in the limit as time to infinity [13][14][15][16]20]. It is remarkable that, in Theorem 1.5, in conjunction with Proposition 1.3(ii), this function turns out also to govern the large deviations for the mean of the normalised occupation measures under the symmetrised measure µ (sym) ,N , in the limit of large number of motions.…”

Section: An Important Special Casementioning

confidence: 99%

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Adams

König

2007

Probab. Theory Relat. Fields

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Consider a large system of N Brownian motions in R d with some nondegenerate initial measure on some fixed time interval [0, β] with symmetrised initialterminal condition. That is, for any i, the terminal location of the i-th motion is affixed to the initial point of the σ (i)-th motion, where σ is a uniformly distributed random permutation of 1, . . . , N . Such systems play an important role in quantum physics in the description of Boson systems at positive temperature 1/β. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of e −βH N , where H N is an N -particle Hamilton operator in a trap.

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“…Moreover, since EIj is finite it is clear that p(IIx), being an isolated root of the characteristic equation for the matrix II, is differentiable with respect to A (see [14] for details). Therefore, Theorem 2.3.1 may be applied to complete the proof.…”

Section: Then I() Is a Good Convex Rate Function Which Controls Tmentioning

confidence: 99%

“…The credit for the extension of Cramer's theorem to the dependent case should definitely go to Ga.rtner [14] who considered the case in which DA = IR d . Ellis [10] extended this result to the steep set up and the formulation of section 2.3 is nothing but an embellishment of his results.…”

Section: Chaptermentioning

confidence: 99%

Large Deviations and Applications: The Finite Dimensional Case

Dembo¹,

Zeitouni²

1991

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Preface and WarningThese notes, which form the first chapter of a forthcoming book, are intended to serve as lecture notes on the topic of large deviations and applications for students whose background and interests are in applications which involve finite dimensional spaces. Although narrow in their scope, these notes present a good deal of the methods available for more general situations. A glaring omission is the method of sub-additivity, which will be discussed in another chapter in the book. Another deficiency of these notes is the sketchy bibliography and historical. We hope to correct this in the book.

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On Large Deviations from the Invariant Measure

Cited by 280 publications

References 8 publications

Large deviations behavior of counting processes and their inverses

Large deviations behavior of counting processes and their inverses

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Large Deviations and Applications: The Finite Dimensional Case

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