System Theory

Download Large Deviations Techniques and Applications by Amir Dembo, Ofer Zeitouni PDF

By Amir Dembo, Ofer Zeitouni

The concept of huge deviations bargains with the evaluate, for a kin of likelihood measures parameterized through a true valued variable, of the chances of occasions which decay exponentially within the parameter. initially constructed within the context of statistical mechanics and of (random) dynamical platforms, it proved to be a strong software within the research of platforms the place the mixed results of random perturbations bring about a habit considerably various from the noiseless case. the quantity enhances the vital parts of this conception with chosen functions in communique and keep watch over platforms, bio-molecular series research, speculation trying out difficulties in records, and the Gibbs conditioning precept in statistical mechanics.

Starting with the definition of the big deviation precept (LDP), the authors supply an summary of huge deviation theorems in ${{\rm I\!R}}^d$ by means of their software. In a extra summary setup the place the underlying variables take values in a topological area, the authors supply a suite of tools aimed toward developing the LDP, akin to adjustments of the LDP, kin among the LDP and Laplace's procedure for the evaluate for exponential integrals, houses of the LDP in topological vector areas, and the habit of the LDP lower than projective limits. They then flip to the learn of the LDP for the pattern paths of sure stochastic procedures and the applying of such LDP's to the matter of the go out of randomly perturbed suggestions of differential equations from the area of appeal of sturdy equilibria. They finish with the LDP for the empirical degree of (discrete time) random strategies: Sanov's theorem for the empirical degree of an i.i.d. pattern, its extensions to Markov tactics and combining sequences and their software.

The current delicate conceal variation is a corrected printing of the 1998 edition.

Amir Dembo is a Professor of arithmetic and of facts at Stanford college. Ofer Zeitouni is a Professor of arithmetic on the Weizmann Institute of technology and on the college of Minnesota.

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Let Hρ =[−ρ, ρ]d . 33) j=1 where μjn , j = 1, . . , d are the laws of the coordinates of the random vecn j tor Sˆn , namely, the laws governing n1 i=1 Xi . 2 Crame j where Λ∗j denote the Fenchel–Legendre transform of log E[eλX1 ], j = 1, . . , d. 20, Λ∗j (x) → ∞ when |x| → ∞. 33) and considering the limits, first as n → ∞ and then as ρ → ∞, one obtains the identity lim lim sup ρ→∞ n→∞ 1 log μn (Hρc ) = −∞ . n Consequently, {μn } is an exponentially tight sequence of probability measures, since the hypercubes Hρ are compact.

Prove that H(ν|·) is a continuous function for every fixed ν ∈ M1 (Σ), and use it for proving the large deviations lower bound and the lower semicontinuity of the rate function I(·). d. random variables is proved. d. 1. Without loss of generality, it is further assumed that Σ = Σμ and that f (a1 ) < f (a2 ) < · · · < f (a|Σ| ). Cram´er’s theorem deals with the LDP associated with the real-valued random variables Sˆn . 3). Note that in the case considered here, the random variables Sˆn assume values in the compact interval K =[f (a1 ), f (a|Σ| )].

D. Poisson(1) random variables. 19 Let N (n) be a sequence of nonnegative integer-valued random variables such that the limit Λ(λ) = limn→∞ n1 log E[eλN (n) ] exists, and o . d. random variables, independent of {N (n)}, 0 ∈ DΛ 52 2. LDP for Finite Dimensional Spaces with finite logarithmic moment generating function ΛX . Let μn denote the law of N (n) 1 Xj . Zn = n j=1 (a) Prove that if the convex function Λ(·) is essentially smooth and lower semicontinuous, then so is Λ(ΛX (·)), and moreover, Λ(ΛX (·)) is finite in some ball around the origin.

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