= Solution
{c}
The long-time <Spitzer winding theorem> states that, for the continuous unwrapped angle $\Theta_t$ of planar <Brownian motion> started at $1$,
$$
\boxed{\frac{2\Theta_t}{\log t}\ \xrightarrow[t\to\infty]{\mathrm d}\ C,\qquad
f_C(x)=\frac1{\pi(1+x^2)}.}
$$
Here $C$ has the standard symmetric <Cauchy distribution>. The notation means <convergence in distribution>: for every real $x$,
$$
\lim_{t\to\infty}\mathbb P\left(\frac{2\Theta_t}{\log t}\leq x\right)
=\frac12+\frac1\pi\arctan x.
$$
Equivalently, expectations converge for every bounded continuous test function. This is a statement about the laws at large deterministic times, without asserting almost-sure convergence of sample paths. The angle is unwrapped; a principal argument is bounded and would instead give zero after division by $\log t$.
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