A concentrated Cauchy distribution with finite moments

Abstract

The Cauchy distribution

$$\mathfrak {C}(a,b)(x)=\frac{1}{\pi b(1+(\frac{x-a}{b})^2)},\quad -\infty < x <\infty,$$

with a,b real, b>0, has no moments (expected value, variance, etc.), because the defining integrals diverge. An obvious way to “concentrate” the Cauchy distribution, in order to get finite moments, is by truncation, restricting it to a finite domain. An alternative, suggested by an elementary problem in mechanics, is the distribution

$${\mathfrak {C}}_g(a,b)(x)=\frac{\sqrt{1+2 b g}}{\pi b (1+(\frac{x-a}{b})^2)\sqrt{1-2 b g(\frac{x-a}{b})^2}},\quad a-\sqrt{\frac{b}{2g}}<x<a+\sqrt{\frac{b}{2g}},$$

with a,b as above and a third parameter g≥0. It has the Cauchy distribution C(a,b) as the special case with g=0, and for any g>0, ℭ _g (a,b) has finite moments of all orders, while keeping the useful “fat tails” property of ℭ(a,b).