Let $X_0,X_1,\dots$ be a sequence of i.i.d. random variables, with $\mathbb{E}\big[|X_i|\big]=\infty$. Show that: $$ 1.\;\limsup \left(\frac{|X_n|}{n}\right)=\infty\qquad 2.\; \limsup \left(\frac{|X_1+\cdots+X_n|}{n}\right)=\infty$$ almost surely.
For the $1.$ I took arbitrary $R>0$ then: $$\begin{aligned}\infty=\frac{1}{R}\,\mathbb{E}\big[|X_1|\big] &=\frac{1}{R}\int_{\mathbb{R}_{\geq 0}}\mathbb{P}(|X_1|\geq x)\,\mathrm{d}x \\ &=\int_{\mathbb{R}_{\geq 0}}\mathbb{P}(|X_1|\geq Rx)\,\mathrm{d}x \quad (x\mapsto Rx)\\ & =\sum_n\int_n^{n+1}\mathbb{P}(|X_1|\geq Rx)\,\mathrm{d}x\\ &\leq\sum_n\int_n^{n+1}\mathbb{P}(|X_1|\geq Rn)\,\mathrm{d}x\quad \big(\mathbb{P}(|X|>a)\geq\mathbb{P}(|X|>b)\;\text{if}\;a<b\big)\\ &=\sum_n\mathbb{P}(|X_1|\geq Rn)\\& =\sum_n\mathbb{P}(|X_n|\geq Rn)\quad (X_1\sim X_n\;\text{for all}\;n)\end{aligned}$$ We can now use Borel-Cantelli's second lemma to infer $\limsup_n |X_n|/n\geq R$ almost surely, and since $R$ was arbitrary $\limsup |X_n|/n=\infty$ almost surely.
- Does my proof work?
- Any hints on the second part? It looks like the strong law, but the modulus puts it on the "wrong" side of the triangle inequality, so I can't apply it.