Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 4

A standard Brownian motion $(B_t{)}_{t\ge 0}$ satisfies $B_0=0$ almost surely, has almost surely continuous sample paths, and has independent stationary increments such that

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The variable $Z={\int }_0^1{B}_{t}dt$ is the $L^2$ limit of Riemann sums of the Gaussian process $B$, so it is a Gaussian random variable. Its mean is zero, and Fubini's theorem with $\mathbb{E}[B_s{B}_t]=\min (s,t)$ givesTherefore $Z\sim N(0,1/3)$.

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Let $X_1,X_2,\dots$ be independent and identically distributed random variables with mean zero and variance one, and let $S_k={\sum }_{j=1}^k{X}_j$. Define the linearly interpolated processThe Donsker invariance principle, also called the functional central limit theorem, states that $W_n$ converges weakly in the space $C[0,1]$ with the uniform norm to standard Brownian motion.

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The evaluation map $f\mapsto f(1)$ is continuous on $C[0,1]$ in the uniform norm. The continuous mapping theorem applied to part (c) therefore givesThis recovers the central limit theorem from the functional version.

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Reversing the finite summations givesThis is a Riemann sum for the continuous functional $f\mapsto {\int }_0^{1}f(t)dt$ evaluated at the interpolated random walk; the interpolation error tends to zero in probability. The continuous mapping theorem and part (b) yield

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