By Oliver T Johnson
ISBN-10: 1860944736
ISBN-13: 9781860944734
This publication offers a complete description of a brand new approach to proving the principal restrict theorem, by utilizing it seems that unrelated effects from info idea. It offers a easy advent to the innovations of entropy and Fisher info, and collects jointly regular effects bearing on their behaviour. It brings jointly effects from a couple of learn papers in addition to unpublished fabric, displaying how the concepts can provide a unified view of restrict theorems.
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Additional resources for Information Theory and the Central Limit
Example text
IID and with finite variance g 2 , define the normalised sum Un = (C,"=, Xi)/@. 4 Given X I ,X 2 , . . IID and with finite variance u 2 , define the normalised sum U, = (CZ,Xi)/@. If Xi have finite Poincare' constant R, then writing D(Un) for D(Unl14): 2R D(un) 2R D ( X ) 5 -D(x) + (2R n - l)a2 nu2 for all n. , 20031 has also considered the rate of convergence of these quantities. Their paper obtains similar results, but by a very different method, involving transportation costs and a variational characterisation of Fisher information.
56) 2 (T1 (PE (d(Y1) - P I 2 + (1 - P)E (d(YZ) -d ) I where 7= (1 - EY, f (Yl + v). Proof. (z. gz(Yz)) P2(YZ)l = EY, [(f(Yl+ ). 59) and show that we can control their norms. 60) (f(. (Yl) I E (f(Y1 + Yz)- Sl(Y1)- g2(Yz))2 J(yz). 67), 0 we deduce the result. Hence we see that if the function of the sum f(Y1+Yz)is close t o the sum of the functions g(Y1)+g(Yz), then g has a derivative that is close to constant. Now, we expect that this means that g itself is close to linear, which we can formally establish with the use of Poincark constants (see Appendix B).
118) substituting flu and m V for U and V respectively, we recover the second result. r(W - + u) (1 - P ) P V ( V ) = PW(W), for all u , w . 119) + Integrating with respect to v , -Plogp(w - u) (1 - P)logq(v) = u ( r ’ ( w ) / r ( w ) ) c ( w ) , Setting w = 0, we deduce that C ( W ) and p w ( w ) are differentiable. Differentiating with respect t o w and setting w = 0, we see that p ’ ( - u ) / p ( - u ) is linear in v , and hence p is a normal density. 0 + This result is a powerful one: it allows us to prove that the Fisher information decreases ‘on average’ when we take convolutions.
Information Theory and the Central Limit by Oliver T Johnson
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