A theorem about probabilities of large deviations with an application to queuing theory

Plachky, Detlef; Steinebach, Josef

doi:10.1007/bf02017929

Cited by 91 publications

(64 citation statements)

References 4 publications

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“…Using that ln ( f n ) dρ t 0 = −P (t 0 ) and that the function s → P(t 0 − s) − P(t 0 ) is real analytic and strictly convex on (t 0 − t + , t 0 − t − ) by Theorem D, we obtain by the theorem in p. 343 of [PS75] that lim n→+∞ 1 n ln ω n 1 n ln f n > ln f n dρ t 0 + ε…”

Section: Theorem Dmentioning

confidence: 88%

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Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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Abstract:We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln f , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions.

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Section: Theorem Dmentioning

confidence: 88%

“…We will apply the theorem in p. 343 of [PS75] to the space J := +∞ n=1 J ( f ) endowed with the probability measure P := +∞ n=1 ω n . Furthermore for each n ≥ 1 we take the random variable W n : J → R as W n +∞ j=1 x j := ln ( f n ) (x n ) .…”

Section: Theorem Dmentioning

confidence: 99%

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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“…Therefore we have reduced the statements of the theorem aboutṠ W n to proving (18). By definition we have…”

Section: Proof Of Theoremmentioning

confidence: 99%

Testing the irreversibility of a Gibbsian process via hitting and return times

Chazottes¹,

Redig²

2005

Nonlinearity

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“…(Note that if ¡fdp = 0, then 1(a) = a2/2<r2 + 0(a3) in the limit a -» 0.) Here we used [Plachky/Steinebach, 1975]. A good discussion of the Laplacetransform technique can be found in [Cox/Griffeaths, 1984].…”

Section: Applicationsmentioning

confidence: 99%