[Math] Prove: Almost sure convergence of random variables with Borel Cantelli lemma

borel-cantelli-lemmasprobability theory

Let $X_n$ be a sequence of random variables with $X_n<\infty$ almost sure for all $n\in \mathbb N$. Show that there are constants $c_n\rightarrow \infty$ such that $\frac{X_n}{c_n}\rightarrow 0$

I guess I have to use the Borel Cantelli Lemma, right?

Best Answer

Hints:

  1. Show that for $n \in \mathbb{N}$ and $\varepsilon>0$ there exists $R>0$ such that $$\mathbb{P}(|X_n|>R) < \varepsilon.$$ To this end, use that $X_n< \infty$ almost surely.
  2. By step 1, we can choose $R_n$ such that $$\mathbb{P}(|X_n| >R_n) \leq 2^{-n}.$$ Show for $c_n := n \cdot R_n$ that $$\sum_{n \geq 1} \mathbb{P} \left(|X_n| \geq \frac{c_n}{n} \right) < \infty.$$ Using the Borel-Cantelli theotem, conclude that $$ \frac{|X_n|}{c_n} \leq \frac{1}{n}$$ for $n$ sufficiently large.
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