Past exam of the random matrix theory course of the University of Oxford / 2025 / 3 / c

The characteristic function of $B(t)\sim N(0,t)$ gives $\mathbb{E}{e}^{2iB(t)}=e^{-2t}$. Taking real parts and using ${\sin }^{2}b=(1-\cos 2b)/2$,There is also a direct check from the stochastic differential equation. Its Itô integral has zero expectation because its integrand is bounded. Thus $m(t)=\mathbb{E}\Delta (t)$ satisfies $m^{\prime }(t)=1-2m(t)$ and $m(0)=0$, whose solution is the same expression.

Interpret the source's missing variable in the limit arrow as $t\to \infty$. We prove convergence in distribution by showing that the angle of the Brownian motion becomes uniform on the circle group.

Let $Z_t=e^{iB(t)}$. For every integer $m$, its angular Fourier coefficients satisfyThese are precisely the Fourier coefficients of the uniform distribution on the unit circle. Thus expectations of every trigonometric polynomial converge to its uniform angular average. By the Stone-Weierstrass theorem, trigonometric polynomials approximate continuous functions on the unit circle uniformly. Given an approximation error $\epsilon$, the discrepancy of expectations is at most the polynomial discrepancy plus $2\epsilon$. Taking $t\to \infty$ and then $\epsilon \to 0$ proves weak convergence of probability measures to uniform angular measure.

The function $z\mapsto (\mathrm{Im}z{)}^2$ is continuous on the unit circle, so the continuous mapping theorem and part (a)(ii) implyEquivalently, the limiting arcsine distribution has cumulative distribution function $2\arcsin \sqrt{x}/\pi$ for $0\le x\le 1$, and probability density function $1/(\pi \sqrt{x(1-x)})$ for $0<x<1$. This proves the limiting law rather than inferring it merely from the limiting mean.