Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 27J / a / Solution

The strong law of large numbers states that for independent identically distributed real variables with $\mathbb{E}|X_1|<\infty$, the sample average converges almost surely to $\mathbb{E}{X}_1$. The central limit theorem states that if additionally $0<\mathrm{Var}{X}_1={\sigma }^2<\infty$, thenThe vector form uses finite second moments and gives a centred multivariate Gaussian with the covariance matrix of one observation.