Examples of convergence in distribution not implying convergence in moments

Question

It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.

Let $\{X_n\}$ be a sequence of random variables that converge in distribution to $X$. I would like to ask two examples as follows.

1. An example such that $\mathbb{E}[X_n]$ does not convergence to $\mathbb{E}[X]$.
2. An example such that $\mathbb{E}[X_n^k]$ does not convergence to $\mathbb{E}[X^k]$ for all $k = 1, 2,\ldots$. That is, the sequence does not converge in all the moments, not just a or a few fixed moments. Suppose that all their moments exist.

Answer 1

Let $P(X_{n} = n) = \frac{1}{n}$ and $P(X_{n} = 0) = 1 - \frac{1}{n}$. Then $X_{n}$ converges in distribution to $X=0$, but $\mathbb{E}(X_{n}^{k}) = \frac{1}{n} n^{k} + 0 = n^{k-1} \not\xrightarrow{n\to\infty} 0$ for each $k \in \mathbb{N}$.

UPDATE: If you prefer continuous random variables on $\mathbb{R}$, you can "smoothen out" the previous example:

Let $X \sim N(0,1)$ and $X_{n}$ have the probability density $$ \rho_{X_{n}}(x) = \frac{1}{n} \cdot \frac{1}{\sqrt{2\pi}}\, \exp(-(x-n)^2/2) + \left(1-\frac{1}{n}\right) \cdot \frac{1}{\sqrt{2\pi}}\, \exp(-x^2/2). $$ Then $X_{n} \stackrel{\mathrm{d}}{\to} X$.

Now denote the moments of $X$ by $A_{k} := \mathbb{E}(X^{k})$ and, for $m \geq 0$, let $A_{k,m} := \mathbb{E}((X+m)^{k}) \geq A_{k} + m^k$. It follows for each $k\in\mathbb{N}$: \begin{align*} \mathbb{E}(X_{n}^{k}) &= \int_{\mathbb{R}} x^{k} \rho_{X_{n}}(x)\, \mathrm dx \\ &= \frac{1}{n} A_{k,n} + \left(1-\frac{1}{n}\right) A_{k} \\ &\geq \frac{1}{n} (A_{k}+n^{k}) + \left(1-\frac{1}{n}\right) A_{k} \\ &= n^{k-1} + A_{k} \\ &\not\xrightarrow{n\to\infty} A_{k}. \end{align*}