Spitzer winding theorem ($2{\Theta }_t/\log t\Rightarrow C$)

For planar Brownian motion started at $1$, let ${\Theta }_t$ be its continuous unwrapped angle, initialized at zero. Then $2{\Theta }_t/\log t$ converges in distribution to the standard symmetric Cauchy distribution, with density $1/[\pi (1+x^2)]$. The angle records all completed turns rather than a bounded principal argument. The assertion is a convergence in distribution at deterministic times; it does not assert convergence of sample paths. A precise statement appears as Theorem 7.31 in Mörters and Peres, Brownian Motion.