Can one prove for a sequence of positive random variable $X\_{n}$ such that $\lim\_{n\to \infty}E[x\_{n}] = 0$ and $\lim\_{n\to \infty}E[x\_{n}x\_{n}]= 0$ all the cumulants go to zero once $n\to \infty$ ?



Cumulants of a sequence of variables with zero mean and variance
Counter example: probability distribution of $X\_n$ given by
$$P\_n(x)=\frac{3n}{(1+nx)^4},\;\;x\geq 0,$$
properly normalized to unity. Then $\mathbb{E}[X\_n]=1/2n$ and $\mathbb{E}[X\_n^2]=1/n^2$ both vanish in the limit $n\rightarrow\infty$, but higher moments and cumulants diverge.


Alternatively, for an example where the moments do not diverge for finite $n$, let $X\_n$ take the value $1/n$ with probability $1-1/n$ and the value $n^{1/3}$ with probability $1/n$. Then $\mathbb{E}[X\_n]$ and $\mathbb{E}[X\_n]^2$ both vanish in the limit $n\rightarrow\infty$, while $\mathbb{E}[X\_n^3]$ tends to unity.


