Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 33 / 6 / b / Solution

Let $X_i$ be independent and identically distributed random variables with mean $0$ and variance ${\sigma }^2\in (0,\infty )$, and put $S_0=0$, $S_k={\sum }_{i=1}^k{X}_i$. Define the polygonal interpolationwith the endpoint $W_n(1)=S_n/(\sigma \sqrt{n})$. The Donsker invariance principle stateswhere $B$ is standard one-dimensional Brownian motion and the arrow denotes weak convergence of probability measures on this path space. Equivalently, the step interpolation converges in the Skorokhod space with its $J_1$ topology. The continuous interpolation version is sufficient for the maximum in part (c).