# Tagged Questions

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### How does $\mathcal{L}^1$-convergence of a series of $\mathcal{L}^1$ random variables imply that $\sup_{n \in \mathbb{N}} \mathbb{E}[|X_n|] < \infty$?

Let $(X_n)_{n \in \mathbb{N}}$ be a series of random variables with $\forall i: X_i \in \mathcal{L}^1(\Omega, \mathfrak{F}, P)$ and $X_n \rightarrow^{\mathcal{L}^1}X$. How do I show then, that ...
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### Convergence of Random Variables in mean

If $$E[|X_n-X|^r]\rightarrow0$$ prove that $$E|X_n^r|\rightarrow E|X^r|$$ for every $r\ge 1$ This is the very notation used. I believe it should be: $$E[|X_n|^r]\rightarrow E[|X|]^r$$ Attempt I ...
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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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### Weak convergence in the Skorohod space $D([0, T])$ $\forall T$ implies Weak convergence in $D([0, \infty))$?

Assume that $Z_n$ are random variables taking value in the Skorohod space $D([0, \infty),Y)$ (endowed with its usual Skorohod topology) of right-continuous functions $[0, \infty) \to Y$, where $Y$ is ...
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### Convergence in Probabiliy

$X_n$ converges to $0$ in probability and a sequence of constants $|c_n|$ diverges to infinity. Can someone please help me prove that $X_n - c_n$ does not converge to $0$. (I am totally blank as to ...
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