# Simulating a Cauchy process

My question is how to I simulate sample paths from a Cauchy process? I know this can be done using two Brownian motions, but I am trying to do it from the basics.

It's known that if we have a Levy process $$\{X_t\}$$ with Levy triplet $$(0,0,\nu)$$ then the measure $$\nu$$ determines the jump intensity such that if $$\lambda:=\nu(\mathbb{R})<\infty$$ then the probability measure $$\nu_1=\nu/\lambda$$ determines the distribution of jump sizes; furthermore $$X_t=\sum_{j=1}^{N_t}V_j ,$$ where $$\{N_t\}$$ is a Poisson process with intensity $$\lambda$$ and the $$V_j$$ are iid random variables that are distributed according to $$\nu_1$$.

However, in the case of a Cauchy process we have the triplet $$(0,0,W)$$, where $$W(dx)=\frac{dx}{\pi x^2}.$$ And here $$W(\mathbb{R})$$ is not finite. Is there a way to side-step this?

Thanks in advance.