Let \(\C\) denote the set of complex numbers \(z = x + iy\) with the topology of \(\R^2\), and let \(\cdot\) denote ordinary complex multiplication. Let \(\C_0 = \C - \{0\}\) and let \[S = \{z \in \C: |z| \gt 1\} \cup\{1\}\] both with the relative topology.
\((S,\,\cdot)\) is a positive sub-semigroup of the abelian group \((\C_0,\,\cdot)\), The induced partial order is given by \[z \prec w \text{ if and only if } |z| \lt |w|\]
\(z \prec w\) if and only if there exists \(u \in S\), \(u \neq 1\), such that \(z u = w\). But this occurs if and only if \(|u| = |w / z| = |w| / |z| \gt 1\).
The measure \(\lambda\) defined by \[d\lambda(z) = \frac{1}{|z|^2} dx dy = \frac{1}{x^2 + y^2} dx dy\] is left-invariant on the group \((\C - \{0\},\,\cdot)\). Hence \(\lambda\) restricted to \(S\) is left-invariant on \(S\).
Let \(f \colon S \rightarrow \R\) be measurable. It suffices to show that for \(w \in \C_0\), \[\int_{\C_0} f(wz) d\lambda(z) = \int_{\C_0} f(z) d\lambda(z)\] Let \(z = r e^{i \theta}\) where \((r,\,\theta)\) denote ordinary polar coordinates in \(\C\), \(0 \leq \theta \lt 2 \pi\). Then \[d\lambda(z) = \frac{1}{r^2} dx dy = \frac{1}{r^2} r dr d\theta = \frac{1}{r}dr d\theta\] Let \(w = \rho e^{i \phi}\). Then \[\int_{\C_0} f(wz) d\lambda(z) = \int_0^{2\pi} \int_0^\infty f(r \rho e^{i(\theta + \phi)}) \frac{1}{r} dr d\theta\] Now let \(\hat{r} = r \rho\), \(\hat{\theta} = \theta + \phi\). Then \(dr = (1 / \rho) d\hat{r}\) and \(d\theta = d \hat{\theta}\) so \[\int_{\C_0} f(wz) d\lambda(z) = \int_0^{2 \pi} \int_0 ^\infty f \left(\hat{r} e^{i \hat{\theta}}\right) \frac{1}{\hat{r}} d\hat{r} d\theta = \int_{\C_0} f(z) d\lambda(z)\]
\(F : S \to (0, 1]\) is the reliabililty function of a memoryless distribution on \((S, \, \cdot)\) if and only if \(F(z) = |z|^{-\beta}\) for some \(\beta \gt 0\). The corresponding density function (with respect to \(\mu\)) is \[f(z) = \frac{\beta}{2 \pi} |z|^{-\beta}, \quad z \in S\]
Suppose that \(F : S \to (0, 1]\) satisfies \(F(zw) = F(z)F(w)\) for \(z, \, w \in S\). If \(x, \, y \in [1, \, \infty)\) then \(F(xy) = F(x)F(y)\); that is, \(F\) is a homomorphism restricted to the multiplicative semigroup \([1, \, \infty)\). Hence there exists \(\beta > 0\) such that \(F(x) = x^{-\beta}\) for \(x \in [1, \, \infty)\). If \(x > 1\) then \([F(-x)]^2 = F(x^2) = x^{-2 \beta}\) so \(F(-x) = x^{-\beta}\) and therefore \(F(x) = |x|^{-\beta}\) for \(x \in (-\infty,\,1) \cup [1,\,\infty)\). Next, if \(r > 1\), \(m \in \N\) and \(n \in \N_{+}\) then \[[F(r e^{i \pi m / n})]^n = F[(r e^{i \pi m / n})^n] = F(r^n e^{i m \pi}) = r^{-n \beta}\] Hence \(F(r e^{i m \pi / n}) = r^{-\beta}\). By continuity, \(F(r e^{i \theta}) = r^{-\beta}\) for \(r > 1\) and any \(\theta\). Next note that \begin{align*} \int_S F(z) \, d\lambda(z) &= \int_S |z|^{-\beta} \frac{1}{|z|^2} \, dx \, dy \\ &= \int_0^{2 \pi} \int_1^\infty r^{-(\beta + 2)} r \, dr \, d\theta = \int_0^{2 \pi} \int_1^\infty r^{-(\beta + 1)} \, dr \, d\theta = \frac{2 \pi}{\beta} \end{align*} From the basic existence theorem, \(F\) is the reliability function of an memoryless distribution on \((S, \, \cdot)\) and that \(f\) given above is the corresponding density function (with respect to the left-invariant measure \(\lambda\)).
Suppose that \(Z\) has the memoryless distribution on \((S, \, \cdot)\) with parameter \(\beta \gt 0\), as specified in the previous theorem. Then the density function with respect to Lebesgue measure is \[z \mapsto \frac{\beta}{2 \pi} |z|^{-(\beta + 2)}\] In terms of polar coordinates, the density is \[(r, \theta) \mapsto \frac{\beta}{2 \pi} r^{-(\beta + 1)}\] It follows that \(Z = R e^{i \Theta}\) where \(R > 1\) and \(0 \lt \Theta \lt 2 \pi\); \(R\) has density \(r \mapsto \beta r^{-(\beta + 1)}\) (with respect to Lebesgue measure); \(\Theta\) is uniformly distributed on \((0,\,2 \pi)\); and \(R\) and \(\Theta\) are independent.