= Spitzer winding theorem
{c}
{title2=$2\Theta_t/\log t\Rightarrow C$}
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 https://www.stat.berkeley.edu/~aldous/205B/bmbook.pdf[Mörters and Peres, Brownian Motion].
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