By Jacques Azema
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Additional resources for Seminaire De Probabilites XVIII 1982/83
Indeed, Tribus in [Levine and Tribus, 19791 reports a conversation where von Neumann suggested to Shannon that he should use the same name: 18 Information Theory and the Central Lzmit Theorem You should call it ‘entropy’ and for two reasons; first, the function is already in use in thermodynamics under that name; second, and more importantly, most people don’t know what entropy really is, and if you use the word ‘entropy’ in an argument you will win every time. In the study of statistical physics, we contrast macrostates (properties of large numbers of particles, such as temperature and pressure) and microstates (properties of individual molecules, such as position and momentum).
Whilst it might sound surprising to refer t o such a well-known and long-established principle in 20 Information Theory and the Central Limit Theorem this way, there remains a certain amount of argument about it. Depending on the author, the Second Law appears t o be treated as anything from something so obvious as not t o require discussion, to something that might not even be true. A recent discussion of the history and status of the Second Law is provided by [Uffink, 20011. He states that: Even deliberate attempts at careful forniulation of the Second Law sometimes end up in a paradox.
1 Let 4 be the N ( 0 , l ) density. Given IID random variables X l , X z , . . with densities and variance n2, let gn represent the density of U, = Xi) The relative entropy converges to zero: (c:=l /m. 55) Convergence in relative entropy 43 if and only if D(gnI14)is finite for some n. Proof. 1 as a starting point, using a uniform integrability argument t o show that the Fisher information converges t o l / ( l + ~ Convergence ). in relative entropy follows using de Bruijn’s identity (Theorem C .