Cumulants of a sequence of variables with zero mean and variance

Question

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 $n\to \infty$ ?

Answer 1

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.

Answer 2

Take a bernoulli variable with weight $\log^{-4} n$ at $-\log n$ and weight $1-\log^{-4} n$ at $\log n/(\log^4 n-1)$.

Then the mean is 0, the variance is $O(\log^{-2} n)$, the 4th moment converges to 1 and the 6th and 8th moment go to $+\infty$ while the 5th and 7th moments go to $-\infty$. Probably later moments behave the same.

The fourth cumulant tends to 1 and the sixth cumulant goes to infinity.