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Planar Brownian motion started at $(-x,0)$ exits the left half-plane with ordinate distributed as $x$ times a standard Cauchy random variable. Conditional on the independent horizontal hitting time, the ordinate is normal with variance equal to that time. The inverse-square Gaussian law of Brownian first passage turns this into a ratio of independent standard normals. Its exit density is $x/(\pi (x^2+y^2))$.

The Laplace kernel on ${\mathbb{R}}^d$ is a translation-invariant positive-semidefinite kernel. Independent standard Cauchy random variables give its Random Fourier feature representation through their characteristic function $e^{-|t|}$.

For events $A_n$, their limsup is the event that infinitely many occur. The first Borel-Cantelli lemma states that ${\sum }_n\mathbb{P}(A_n)<\infty$ implies $\mathbb{P}(\mathrm{limsup}{A}_n)=0$, without independence. Indeedand intersecting these decreasing tail-unions proves the claim.

The second Borel-Cantelli lemma states that independent events with ${\sum }_n\mathbb{P}(A_n)=\infty$ satisfy $\mathbb{P}(\mathrm{limsup}{A}_n)=1$. For each fixed $N$,Thus the probability that no event occurs after $N$ is zero; take the countable union over $N$ to exclude finitely many occurrences.

As printed, the requested real logarithm is undefined: independent Cauchy random variables are negative with probability $1/2$, and the second Borel-Cantelli lemma shows that this happens infinitely often almost surely. There is therefore no real-valued sequence $\log X_n/\log n$ as stated. The natural corrected expression uses $\log |X_n|$ (or takes the limsup only over indices with $X_n>0$).

For the absolute-value interpretation,For every $q>1$ the events $|X_n|>n^q$ are summable and hence occur only finitely often; for $0<q\le 1$ they are independent with divergent probability sum and occur infinitely often. Applying the two Borel-Cantelli lemmas to a countable sequence of $q$ approaching one from above and below givesThe one-sided tail $\mathbb{P}(X_n>t)\sim 1/(\pi t)$ gives the same constant $1$ for the positive-subsequence interpretation. It does not repair the undefined logarithms at the remaining indices unless that interpretation is explicitly adopted.

Let $V_1,\dots ,V_d$ be independent standard Cauchy random variables. Their characteristic functions and independence giveTaking real parts yieldsFor each realization of $V$, both products are rank-one positive-semidefinite kernels. Their sum and then their expectation remain positive semidefinite by the closure property of positive-semidefinite kernels. This is a Random Fourier feature representation of the Laplace kernel.

For planar Brownian motion started at radius $\epsilon$ and stopped on first hitting radius one, the net winding count divided by $\log \epsilon$ converges to a standard Cauchy random variable divided by $2\pi$. The complex exponential maps Brownian coordinates in the left half-plane to the punctured unit disk after the conformal Brownian clock. The unwrapped argument at exit is the vertical half-plane exit coordinate, with the Cauchy exit law from a Brownian half-plane. Counting integer turns introduces only a uniformly bounded error in winding units.