Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 26 / 5 / Solution

Use the Donsker invariance principle: for independent and identically distributed random variables of mean zero and variance one, the linearly interpolated processes $W_n$ with $W_n(k/n)=S_k/\sqrt{n}$ converge in distribution to standard Brownian motion in $C[0,1]$ with the uniform topology. No moment beyond the finite second moment is required.

The maximum functional $F(f)={\max }_{0\le t\le 1}f(t)$ is continuous, since $|F(f)-F(g)|\le \Vert f-g{\Vert }_{\infty }$. A linear function on each interpolation interval has its maximum at an endpoint, so $F(W_n)=M_n/\sqrt{n}$. The continuous mapping theorem therefore givesBy the Brownian reflection principle, the limiting random variable has the distribution of $|Z|$, $Z\sim N(0,1)$. Its distribution function is continuous and has no atom at any $x\ge 0$, so convergence in distribution permits passage to these tail probabilities. HenceAt $x=0$ the left side is exactly one for every $n$, because $S_0=0$ is included in the maximum, and the right side is also one.