Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles
نویسندگان
چکیده
We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.
منابع مشابه
The Shannon-McMillan theorem and related results for ergodic quantum spin lattice systems and applications in quantum information theory
The aim of this thesis is to formulate and prove quantum extensions of the famous Shannon-McMillan theorem and its stronger version due to Breiman. In ergodic theory the Shannon-McMillan-Breiman theorem is one of the fundamental limit theorems for classical discrete dynamical systems. It can be interpreted as a special case of the individual ergodic theorem. In this work, we consider spin latti...
متن کاملWeak Gibbs Measures: Speed of Convergence to Entropy, Topological and Geometrical Aspects
Abstract. In this paper we obtain exponential large deviation bounds in the Shannon-McMillan-Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of finite type. We also prove almost sure estimates for the error term in the convergence to entropy given by Shannon-McMillan-Breiman formula for both uniformly and non-uniformly expanding shif...
متن کاملThe dimension of ergodic random sequences
Let μ be a computable ergodic shift-invariant measure over {0, 1} . Providing a constructive proof of Shannon-McMillan-Breiman theorem, V’yugin proved that if x ∈ {0, 1} is Martin-Löf random w.r.t. μ then the strong effective dimension Dim(x) of x equals the entropy of μ. Whether its effective dimension dim(x) also equals the entropy was left as an problem question. In this paper we settle this...
متن کاملMaximum-likelihood estimation for hidden Markov models
Hidden Markov models assume a sequence of random variables to be conditionally independent given a sequence of state variables which forms a Markov chain. Maximum-likelihood estimation for these models can be performed using the EM algorithm. In this paper the consistency of a sequence of maximum-likelihood estimators is proved. Also, the conclusion of the Shannon-McMillan-Breiman theorem on en...
متن کاملA Bilateral Version of the Shannon-McMillan-Breiman Theorem
We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by CT n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tmα ∨ T−1α ∨ . . . ∨ T−nα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ ( −1 n+m ) logμ(C n,m,α(x)) = hμ(T, α), where μ is a T ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017