Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 208 / 1

Solution

Disintegrate

Q

successively as

Q (d y) = Q_{1} (d y_{1}) Q_{2} (d y_{2} ∣ y_{1}) \dots Q_{N} (d y_{N} ∣ y_{< N}) .

For each history

y_{< i}

, choose an optimal coupling of

P_{i}

and

Q_{i} (\cdot ∣ y_{< i})

. Sampling these couplings recursively produces a joint law of

(X, Y)

with

Y \sim Q

. Its conditional

X_{i}

-marginal is always

P_{i}

and is independent of the past, so

X \sim P_{1} \otimes \dots \otimes P_{N} = P

. It is therefore a coupling

π \in Π (P, Q)

Write

c_{i} (y_{< i})

for the conditional expected cost

w (X_{i}, Y_{i})

in the chosen coordinate coupling. The assumed one-coordinate transport-entropy inequality gives

ϕ (c_{i} (y_{< i})) \leq D (Q_{i} (\cdot ∣ y_{< i}) ∥ P_{i}) .

\sum_{i = 1}^{N} ϕ (E_{π} w (X_{i}, Y_{i})) \leq \sum_{i = 1}^{N} E_{Q} ϕ (c_{i} (Y_{< i})) \leq \sum_{i = 1}^{N} E_{Q} D (Q_{i} (\cdot ∣ Y_{< i}) ∥ P_{i}) = D (Q ∥ P) .

Solved by gpt-5.6-sol high.

There are no discussions about this article yet.

There are currently no matching articles.