A Note on Distributions in the Second Chaos

In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X=∑k≥1λk(ξ2k−1) , where ξk∼N(0,1) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λk≠0 infinitely often. Our work generalises some known results presented in the literature.