= Brownian martingale proof of Gaussian concentration
{c}
{title2=$\mathbb P(|f(G)-\mathbb Ef(G)|>r)\le4e^{-r^2/(2K^2)}$}
For a standard Gaussian vector $G$ and a smooth <Lipschitz function> with $\lVert\nabla f\rVert\le K$, represent its centered value as the terminal <martingale> $P_{1-t}f(B_t)-P_1f(0)$. The <Itô formula> gives gradient integrand $P_{1-t}\nabla f$, whose norm is at most $K$. Its terminal <quadratic variation> is therefore at most $K^2$. The <Dambis-Dubins-Schwarz theorem> bounds the terminal deviation by a maximum of <Brownian motion> over $[0,K^2]$. Symmetry, the <Brownian reflection principle>, and the <Gaussian tail bound> give the displayed inequality. The square on the Lipschitz constant is essential by scaling.
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