Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 26K / b / Solution

The observations have finite mean and variance. The strong law of large numbers gives ${\overline{X}}_n\to \mathbb{E}{X}_1$ almost surely, so ${\tilde{\theta }}_n\to \theta$ almost surely and therefore is a consistent estimator. The central limit theorem gives