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Given a sequence of random variables $X_1,X_2,...$ defined on the same probability space $(\Omega,\mathcal{B},P)$.

part 1: Verify that $P(\limsup_{n\to\infty}X_n>x)=0$ if and only if $\lim_{n\to\infty}P(\bigcup_{m=n}^\infty\{X_m>x+\frac1k\})=0$ for all $k\geq 1$

part 2: Suppose $E(X_n)\leq x$ for all $n$ and $\sum_{n=1}^{\infty}\var(X_n)<\infty$.

Prove that $P(\limsup_{n\to\infty}X_n\leq x)=1$

I am guessing that these proofs require the use of Boole and/or Chebychev, but I have always struggled with these.

Help is appreciated!

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  1. The key is to see that $$\lim_{n\to +\infty}P\left(\bigcup_{j\geq n}\{X_j>x+k^{-1}\}\right)=P\left(\bigcap_{n\geq 1}\bigcup_{j\geq n}\{X_j>x+k^{-1}\}\right).$$

    Assume that $P(\limsup_{n\to +\infty}X_n>x)>0$. As the sequence $P(\limsup_{n\to +\infty}X_n>x+k^{—1})$ is increasing, we can find a $k$ such that $P(\limsup_{n\to +\infty}X_n>x+k^{—1})>0$. Let $A:=\{\omega,\limsup_{n\to +\infty}X_n(\omega)>x+k^{—1}\}$. If $\omega\in A$, then we can find $I(\omega)\subset \Bbb N$ infinite such that $X_i(\omega)>x+k^{—1}$ for all $i\in I(\omega)$ hence $\omega\in\bigcap_{n\geq 1}\bigcup_{j\geq n}\{X_j>x+k^{—1}\}$. This proves that the set $\bigcap_{n\geq 1}\bigcup_{j\geq n}\{X_j>x+k^{—1}\}$ has a positive probability.

    Conversely, assume that the latter set has a positive probability for some $k\geq 1$. For $\omega$ in this set we have a $\limsup_{j\to +\infty} X_j(\omega)\geq x+k^{—1}>x$.

  2. By the first question, we have to show that for each $k\geq 1$, we have $P\left(\bigcup_{j\geq n}\{X_j>x+k^{-1}\}\right)\to 0.$ We can write, as $EX_j\leq x$, $$\bigcup_{j\geq n}\{X_j>x+k^{-1}\}\subset \{(X_j-EX_j)^2\geq (x+k^{—1}-EX_j\})^2,$$ hence $$P\left(\bigcup_{j\geq n}\{X_j>x+k^{-1}\}\right)\leq \sum_{j\geq n}\frac 1{(x+k^{—1}-EX_j\})^2}\operatorname{Var}(X_j)=k^2\sum_{j\geq n}\operatorname{Var}(X_j).$$

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