An example of a sequence of random variables with certain properties.

Question

Is there any example of a sequence of random variables $(X_n)_{n=1}^{\infty}$ such that

1)$X_n \geq 0 $,

2)$\mathbb{E}[X_n]<\infty$,

3)$X_n \overset{a.s}{\to} X$ for some integrable random variable $X$

4)$\mathbb{E}[X_n] \to \mathbb{E}[X]<\infty$

but there is no any integrable random variable $Y$ such that $X_n \leq |Y|$?

I found an example in the case where $X_n$ are not positive necessarily but for the case where $X_n \geq 0$ i couldnt find any example.

Thanks in advance!

Answer 1

I found the example, am writing it down if anyone is intrested :

Define $X_n = nI_{(\frac{1}{n+1},\frac{1}{n}]}$ then $X_n$ has the following properties:

1) $X_n\geq 0 $

2) $X_n\overset{as}{\to} X=0$

3) $\mathbb{E}[X_n]=n\biggl(\frac{1}{n}-\frac{1}{n+1}\biggr)=\frac{1}{n+1} \overset{n\to \infty}{\longrightarrow} 0=\mathbb{E}[X]< \infty$

But for $(X_n)_{n=1}^{\infty}$ there is no random variable $Y$ with $\mathbb{E}[\ |Y|\ ]< \infty$ such that $|X_n|\leq |Y|$ since if there exist such $Y$ then its easy to see that for every $n \in \mathbb{N}$

$$\sum_{k=1}^{n}kI_{[\frac{1}{k+1},\frac{1}{k}]} \leq |Y|$$

which implies

$$\sum_{k=1}^{n}\mathbb{E}[\ kI_{(\frac{1}{k+1},\frac{1}{k}]}\ ] \leq \mathbb{E}[\ |Y| \ ]$$

so, $$\sum_{k=1}^{n}\dfrac{1}{k+1}\leq \mathbb{E}[\ |Y|\ ]$$

for every $n \in \mathbb{N}$ , which cant be true , hence there is no such $Y$.