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

Write $m_l=\lfloor N{t}_l\rfloor -\lfloor N{t}_{l-1}\rfloor$ and $\varphi (v)=\mathbb{E}{e}^{iv\xi }$. Since $\mathbb{E}\xi =0$ and $\mathbb{E}{\xi }^2=1$, the characteristic function expansion at zero is $\varphi (v)=1-v^2/2+o(v^2)$. Indeed, $|e^{ix}-1-ix|\le x^2/2$ for real $x$, and $(e^{iv\xi }-1-iv\xi )/v^2\to -{\xi }^2/2$. The dominated convergence theorem, with dominating variable ${\xi }^2/2$, proves the expansion after taking expectations. Initially omit the endpoint interpolation terms. The integer-time increments involve disjoint sets of independent steps, so their joint characteristic function isHere $m_l/N\to t_l-t_{l-1}$. Taking the logarithm near $1$ givesThis proves the claimed limit for the integer-time approximation.

At a fixed time $t$, the omitted endpoint term is $R_N(t)=N^{-1/2}(Nt-\lfloor Nt\rfloor ){\xi }_{\lfloor Nt\rfloor +1}$, and $\mathbb{E}|R_N(t){|}^2\le 1/N$. A fixed linear combination of the errors in the $k$ increments therefore tends to zero in $L^1$, by the triangle inequality and Cauchy-Schwarz inequality, even when two endpoint terms share a step. Since $|e^{ix}-e^{iy}|\le |x-y|$, restoring interpolation changes the joint characteristic function by a quantity tending to zero. HenceThe limit is the joint characteristic function of independent centered normal increments with the indicated variances. By Lévy continuity theorem, the increment vectors converge in distribution to Brownian increment vectors. The value vector is obtained by the fixed invertible triangular map of cumulative sums; thus the continuous mapping theorem gives convergence of $(S_{t_1}^N,\dots ,S_{t_k}^N)$ to $(B_{t_1},\dots ,B_{t_k})$. This proves every required finite-dimensional distribution limit. The finite fourth moment is not needed for this part; it will give tightness in part (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.