Find $X_n$ and $X$, $X_n \xrightarrow{\text{d}} X$ with $X_n$ integrable but $X$ not.

Question

I need to find examples for a series of random variables $\left(X_n\right)_{n \in \Bbb N}$ and $X$, with $X_n \xrightarrow{\text{d}} X$ and one of the following

1. all $X_n$ integrable but $X$ not,
2. all $X_n$ non-integrable , but $X$ is.

I know that integrable means that the integral over the absolute value isn't $\infty$ and that convergence in distribution is equivalent to $F_{X_n}\to F_X$, but I fail to find integrable distribution functions $f_n$ whose integral from $-\infty$ to $x$ converges to $\int_{-\infty}^x f(t)dt$ with $f(t)$ not integrable. (such $f_n$ and $f$ would fulfill the first case I think).

Does anyone have some ideas or tipps on how to find examples? Thanks in advance!

Answer 1

1. Consider $X_n=X\cdot \mathbf 1\left\{-n\leqslant X\leqslant n\right\}$, where $\mathbf 1$ denotes the indicator function and $X$ is a non-integrable random variable (for example, if $X$ has a Cauchy distribution). The convergence is sure hence in probability hence in distribution.

2. Consider $X_n=Y/n$, where $Y$ is a non-integrable random variable. Then $X_n\to 0$ surely hence in probability hence in distribution.