Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 27 / 5 / b

We prove fourth-moment tightness of polygonal random walks uniformly, including intervals shorter than a mesh cell or crossing a mesh boundary. For $s<t$, let $c_j$ be the length of the overlap of $(Ns,Nt)$ with $(j-1,j)$. The interpolated increment isPut $m_4=\mathbb{E}{\xi }^4<\infty$. Expanding the fourth power and using centered independent steps, only the terms with four equal indices or two equal pairs survive. ConsequentlyIf $m_4\le 3$, the last correction is nonpositive. If $m_4>3$, use $\sum c_j^4\le (\sum c_j^2{)}^2$. In either case the expression is at most $\max (3,m_4)N^{-2}(\sum c_j^2{)}^2$. Since $c_j^2\le c_j$,Also $S_0^N=0$ and the paths are continuous. The stated tightness criterion applies with $\gamma =4$, $\alpha =1$ and the displayed constant, so the laws ${\mu }_N$ are tight on $C_0([0,1],\mathbb{R})$ with its uniform topology. Part (a) established all the Brownian finite-dimensional distribution limits. The supplied implication from tightness plus those limits now yields the Donsker invariance principle:where $\mu$ is Wiener measure. No estimate restricted to grid endpoints would suffice by itself for the printed uniform criterion; the overlap coefficients handle every pair of times.