Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 3 / c

Table of contents
- Solution c

Solution

By the stationary distribution property,

X_{1}

and

X_{2}

have the same marginal distribution

π

. Expanding the square gives

\frac{1}{2} E [(f (X_{2}) - f (X_{1}))^{2}] = \frac{1}{2} (2 ⟨ f, f ⟩_{π} - 2 E [f (X_{1}) f (X_{2})]) = ⟨ f, f ⟩_{π} - ⟨ f, K f ⟩_{π} = ⟨ f, (I - K) f ⟩_{π} = E_{K} (f) .

This is the probabilistic representation of the Dirichlet form of a Markov chain.

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