Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 34 / 6 / b / Solution

For a finite-intensity Poisson random measure, independence of counts gives the exponential characteristic function formula. Apply it to the cutoff construction from part (a), including the deterministic compensation, and pass to the L2 space limit of the small jumps. The compensated integrand is $O(y^2)$ at zero and bounded on the large-jump region, so its integral converges absolutely:Symmetry makes the absolutely integrable imaginary part zero. Substituting $z=|u|y$ in the real part and using the specified normalization givesThus $\mathbb{E}{e}^{iu{X}_1}=e^{-|u|}$, and this construction is the standard Cauchy process. To calculate the density, its integrable characteristic function permits Fourier inversion:This is the standard Cauchy distribution. The normalization also gives $c=1/\pi$: integration by parts turns ${\int }_0^{\infty }(1-\cos z)z^{-2}dz$ into ${\int }_0^{\infty }\sin zdz/z=\pi /2$. For completeness, with exponential damping the latter integral equals $\arctan (1/\eta )$, because its derivative in frequency $b$ is ${\int }_0^{\infty }{e}^{-\eta z}\cos (bz)dz=\eta /({\eta }^2+b^2)$ and its value at $b=0$ is zero. Let $\eta \downarrow 0$; integration by parts bounds the undamped and damped tails uniformly by a constant divided by the lower cutoff, justifying the limit.