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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 $\mathbb{E}[X_n]=\frac{n^{4/3}+n-1}{n^2}$ and $\mathbb{E}[X_n]^2=\frac{n^{8/3}+n-1}{n^3}$ both vanish in the limit $n\rightarrow\infty$, while $\mathbb{E}[X_n^3]=\frac{n^4+n-1}{n^4}$ tends to unity.

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.

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.

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 $\mathbb{E}[X_n]=\frac{n^{4/3}+n-1}{n^2}$ and $\mathbb{E}[X_n]^2=\frac{n^{8/3}+n-1}{n^3}$ both vanish in the limit $n\rightarrow\infty$, while $\mathbb{E}[X_n^3]=\frac{n^4+n-1}{n^4}$ tends to unity.

Counter example: probability distribution $$P_n(x)=\frac{3n}{(1+nx)^4},\;\;x\geq 0,$$ 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.

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.