Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 2 / c

Table of contents
- Solution c

Solution

Use the distribution

R

associated with the code in part b. Since

2^{L (x)} = (K R (x))^{- 1}

,

E_{P} 2^{ρ L (X)} = K^{- ρ} \sum_{x} P (x) R (x)^{- ρ} \geq \sum_{x} P (x) R (x)^{- ρ} .

For

α = 1/ (1 + ρ)

, Holder inequality, equivalently the indicated Jensen inequality, gives

\sum_{x} P (x) R (x)^{- ρ} \geq (\sum_{x} P (x)^{α})^{1/ α} .

Therefore

\frac{1}{ρ} lo g_{2} E 2^{ρ L (X_{1}^{n})} \geq \frac{1 + ρ}{ρ} lo g_{2} \sum_{x} P_{n} (x)^{1/ (1 + ρ)} = H_{α} (X_{1}^{n}) .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version