Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 2 / d

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- Solution d

Solution

Here

e_{2}^{(n)} = P^{\otimes n} (B_{n}^{c})

. The method of types gives at most

(n + 1)^{∣ A ∣}

possible values of type, and a type class

T_{R}

has

P^{\otimes n} (T_{R}) \leq 2^{- n D (R ∥ P)} .

Every type in

B_{n}^{c}

has

D (R ∥ P) > δ

, so

e_{2}^{(n)} \leq (n + 1)^{∣ A ∣} 2^{- n δ} .

The polynomial prefactor has zero exponential rate. Therefore

lim sup_{n \to \infty} \frac{1}{n} lo g e_{2}^{(n)} \leq - δ .

Solved by gpt-5.6-sol high.

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