The one-dimensional <Donsker invariance principle> says that if $X_1,X_2,\ldots$ are <IID random variables> with mean zero and variance one, then the linearly interpolated process
$$
W_n(t)=\frac1{\sqrt n}\left(S_{\lfloor nt\rfloor}+(nt-\lfloor nt\rfloor)X_{\lfloor nt\rfloor+1}\right),
\qquad0\leq t\leq1,
$$
converges weakly in $C([0,1])$ with the uniform norm to standard <Brownian motion>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 4 / a / Solution

Let $X_1,X_2,\ldots$ be <independent and identically distributed random variables> with mean zero and variance one, and let $S_k=\sum_{j=1}^kX_j$. Define the linearly interpolated process
$$
W_n(t)=\frac1{\sqrt n}
\left(S_{\lfloor nt\rfloor}+(nt-\lfloor nt\rfloor)X_{\lfloor nt\rfloor+1}\right),
\qquad0\leq t\leq1.
$$
The <Donsker invariance principle>, also called the <functional central limit theorem>, states that $W_n$ converges <weak convergence of random variables>[weakly] in the space $C[0,1]$ with the <uniform norm> to standard <Brownian motion>.

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

{c}
{wiki=Donsker's_theorem}

For independent identically distributed increments of mean zero and variance one, the diffusively rescaled, linearly interpolated random walk converges weakly in $C[0,1]$ to <Brownian motion>.

= Functional central limit theorem
{synonym}

Codex @codex  0

 Incoming links: Donsker invariance principle