# Weak law of large numbers for dampened Cauchy distribution

Say I have a sequence of real independent functions $$X_i$$ on a probability space $$(\Omega, P)$$ such that $$(X_i)_* P$$ gives the probability distribution $$d\alpha := \frac{cdx}{(1+x^2)\log(1+x^2)}$$ (say for $$|x| > 1$$) where $$c$$ is a normalizing constant.

I want to solve an exercise which asks to show $$\frac{S_n}{n} := \frac{X_1 + \dots + X_n}{n}$$ converges in probability to a constant function. Or equivalently that the characteristic function of $$\left(\frac{S_n}{n}\right)_*P$$ converges to that of a probability supported on a point.

This boils down to checking the differentiability at $$0$$ of the characteristic function $$\phi$$ of $$d\alpha$$ which I'm having trouble with (Note, the function $$x$$ is not integrable with respect to $$d\alpha$$). I have

$$\frac{\phi(s) - 1}{s} = \frac{c}{s}\int\limits_{|x| > 1} \frac{e^{isx} - 1}{(1+x^2)\log(1+x^2)}dx$$

and the exercise would follow if I showed the limit as $$s\to 0$$ existed since the characteristic function of $$\left(\frac{S_n}{n}\right)_*P$$ is $$\phi(\frac{\cdot}{n})^n$$.

Can someone please help me evaluate this limit? Normally, pushing the limit inside the integral works but in this case it definitely will not :(

Edit: The original distribution was modified to have support away from $$0$$.

• Surely no such distribution exists because $\int_0^\epsilon\frac{dx}{(1+x^2)\ln(1+x^2)}\sim\int_0^\epsilon\frac{dx}{x^2}=\infty$ for small $\epsilon>0$.
– J.G.
Apr 5 '20 at 16:08
• I see. I'm guessing here, but maybe the exercise wants us to assume that this density is supported away from a neighbourhood of $0$? Can anything be done in this case? For reference, this is Exercise 3.6 in Varadhan's book. I hope I've read correctly.. Apr 5 '20 at 16:33