Vtyjkyuk

I'm working on the following problem and I'm a little stuck

Suppose $X_1,X_2,dots$ are iid.

(a) If $E|X_1|^alpha$ is finite for some $alpha>0$, show that $max_1le kle n |X_k|/n^1/alphato 0$ a.s.

(b) If $EX_1$ is finite and nonzero, show that $max_1le k le n|X_k|/|S_n|to 0$ a.s.

For part a) I'm thinking of using Borel-Cantelli by considering $sum P(max|X_k|^alpha
> epsilon n)$ but I'm having trouble showing it's finite.

For part b) I think you would just use part a) and split up the product into $$fracmaxn^1/alphafracn^1/alpha$$ Then it's just a matter of showing $$fracn^1/alpha$$ converges to something finite, which seems like an SLLN type problem, but not quite. Any advice would be appreciated.

source: Spring 1997

I will use the following two Lemmas, try to prove them on your own:

• If $X_i$ is an iid sequence with $mathbb E|X_1|<infty$, then $X_i/nto 0$ a.s.




• If $a_n$ is a sequence of real numbers with $a_n/nto 0$, then $frac1n max_1le ile na_ito 0.$

Onto your problems:

1. Since $E|X_1|^alpha<infty$, the first Lemma implies $|X_1|^alpha/nto 0$ a.s, then the second lemma implies $(max_1le kle n |X_k|^alpha)/nto 0$ a.s. Conclude by raising each term to the $1/alpha$.




2. You should split it up like $$fracX_kncdot fracn.$$The first fraction converges to $0$ by part (a). Since $n/S_nto 1/EX_1$, you have $n/|S_n|to 1/|EX_1|$.