Outline of the short course •  Part 1: Information theory in a nutshell •  Part 2: The method of types and its relationship with statistics

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Communicated by C. R. Doering Summary. We investigate thermally activated phenomena in micromagnetics using large deviation theory and concepts from stochastic resonance. We give a natural mathe-matical definition of finite-temperature astroids, finite-temperature hysteresis loops, etc. Generically

deviation results for a sum of independent identically distributed random variables and for the joint density of two dependent sums. Then it is applied to a difference approximation to the Helmholtz equation in a random medium. A large deviation result is obtained for the probability density of the decay

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The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic system is observed, the amplitude of the noise perturbing a

A large deviation technique is used to calculate the microcanonical entropy function s(v,m) of the mean-field ϕ 4-model as a function of the potential energy v and the magnetization m. As in the canonical ensemble, a continuous phase transition is found. An analytical expression is obtained

This paper addresses the effects of random perturbations on the long-term behavior of dynamical systems. Large excursions, escape from attractors and the corresponding (first-passage) time scales are predicted probabilistically by two consistent asymptotic theories, the Theory of Large Deviations

Let Sn be a random walk formed by summing i.i.d. integer valued random variables Xi, i ≥ 1: Sn = X1 + · · · + Xn. If the drift EXi is negative, then Sn → − ∞ as n → ∞. If An is the event that Sk ≥ 0 for k = 1,..., n, then P (An) → 0 as n → ∞. In this talk we will consider conditional distributions for the random walk given An. The main result will show that finite dimensional distributions for the random walk given An converge to those for a time homogeneous Markov chain on {0, 1,...}.

are elaborated. Finally, the delay performance of the various approaches are compared through simulations using an example of the uplink OFDMA systems. Index Terms Delay-aware resource control, large deviation theory, Lyapunov stability, Markov decision process, stochastic learning.