Can anyone tell me if the product distribution of (say $n$) IID Cauchy random variables has a tractable form? And if so, what's a good way to go about deriving it? Characteristic functions (ie. Fourier transforms) perhaps? Or is there a simplification of the polynomial form?
Any thoughts are appreciated, thanks.
Best Answer
Let $X,Y$ be iid Cauchy random variables. Then $$ \mathbb P((X,Y)\in A)=\frac{1}{\pi^2}\int_{(x,y)\in A}\frac{dx\ dy}{(1+x^2)(1+y^2)}. $$
Let $U=XY$. It follows that $$ \mathbb P(U\in B)=\frac{1}{\pi^2}\int_{xy\in B}\frac{dx\ dy}{(1+x^2)(1+y^2)}. $$ Let $u=xy$. For fixed $x>0$, we have $y=u/x$ and therefore $dy=du/x$. Since the distribution of $X$ is symmetric, we obtain $$ \begin{align*} \mathbb P(U\in B)&= \frac{2}{\pi^2}\int_{u\in B}\int_{0}^{\infty}\frac{dx\ du/x}{(1+x^2)(1+(u/x)^2)}\\ &=\frac{2}{\pi^2}\int_{u\in B}\int_{0}^{\infty}\frac{x\ dx\ du}{(1+x^2)(x^2+u^2)}\\ &=\frac{2}{\pi^2}\int_{u\in B}\frac{1}{u^2-1}\int_{0}^{\infty}\frac{x}{1+x^2}-\frac{x}{u^2+x^2}\ dx\ du\\ &=\frac{1}{\pi^2}\int_{u\in B}\frac{\log u^2}{u^2-1}\ du. \end{align*} $$
Therefore the density of $U$ is given by $$ \frac{\log u^2}{\pi^2(u^2-1)}. $$