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

Table of contents
- Solution a

Solution

Markov chain Monte Carlo asymptotic variance for a stationary Markov chain and

ψ \in L^{2} (π)

is

σ_{K}^{2} (ψ) = lim_{n \to \infty} n Var (\frac{1}{n} \sum_{j = 1}^{n} ψ (X_{j})) = Var_{π} (ψ) + 2 \sum_{k = 1}^{\infty} Cov_{π} (ψ (X_{0}), ψ (X_{k})),

whenever the limit and series exist. If a reversible Markov chain has positive

L_{0}^{2} (π)

γ

, then the spectral theorem for normal operators on a separable Hilbert space gives

σ_{K}^{2} (ψ) \leq (\frac{2}{γ} - 1) Var_{π} (ψ) .

 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