lecture_slides-Chapter 2.3

# Log n lim pxn h px 0 n1 lim h pxn 1 n1 lim

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Unformatted text preview: 1 lim H (PXn ) = 1. n!1 Thursday, 26 December, 13 An Example • • • • Let X = {1, 2, · · · , }, a countably inﬁnite alphabet. Let X = {1, 2, · · · , }, a countably inﬁnite alphabet. Let PX = {1, 0, 0, · · · }, and let Let PX = {1, 0, 0, · · · }, and let 9 8 > > ⇢ > > < = 1 1 1 ,p ,··· , p , 0, 0, · · · .. PX n = 1 p n > > log n n log n n log n > > : ; | {z } n • As n ! 1, • As n ! 1, X V (PX , PXn ) = X |PX (i) V (PX , PXn ) = i |PX (i) • However, • However, but i H ⇣ 2 PX n ( i ) | = p ! 0. 2 log n PX n ( i ) | = p ! 0. log n ⌘ lim PXn = H (PX ) = 0 n!1 lim H (PXn ) = 1. n!1 lim H (PXn ) = 1. n!1 Thursday, 26 December, 13 An Example • • • • Let X = {1, 2, · · · , }, a countably inﬁnite alphabet....
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