Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 3 / c / Solution

The Markov factorization isDifferentiating its log-likelihood with respect to $\mu$ givesIts coefficients sum to one, so it is unbiased. Direct covariance calculation, equivalently inversion of its Fisher information, givesThus the variance tends to $R$ as $\Delta t/\tau \to 0$, because the observations become perfectly correlated, and to $R/3$ as $\Delta t/\tau \to \infty$, because they become three independent draws.