# Tagged Questions

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### Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
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### Is the product of two independent uniform integrable random variable is uniform integrable?

Is the product of two independent uniform integrable random variable is uniform integrable? What is the role independence plays here?
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### Convergence in probability of the means of a uniformly integrable sequence

Suppose $\{X_n\}$ is a uniformly integrable sequence of independent random variables with zero mean. Prove that $1/n \sum\limits_{i=1}^n X_i \rightarrow0$ in probability. I tried to ...
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### Equivalent condition of uniform integrability of a sequence of random variables

Here's the definition I have for a sequence of random variables to be uniformly integrable: $(1)$ A sequence of random variables $X_1, X_2, \ldots$ is uniformly integrable (U.I.) if for every ...
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### Convergence In $L^{1}$ in the Strong Law of Large Numbers

I'm trying to prove that if $(X_n)_{n\geq 1}$ is uniformly integrable, then $X_n$ almost surely converging to $X$ implies $X_n$ converges to $X$ in $L^{1}$. How is this done? Generally speaking: ...
Let $X$ be a random variable. Are the following three equivalent? $X \in L^1$, i.e. $E |X| < \infty$. $X$ is uniformly integrable. That is, if given $\epsilon>0$, there exists $K\in[0,\infty)$ ...
There are at least 2 places in Wikipedia saying that $X_n$ converges to $X$ in mean in and only if $X_n$ converges to $X$ in probability and $X_n$ is uniformly integrable. See the following link for ...