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$X_1,X_2,\cdots$ are iid with $E(|X_i|^{p})<\infty$ for some real $p\ge 1$ and $E(X_i)=\mu$. I am trying to find the

  1. largest $\alpha>0$ such that $n^{\alpha}\left[\dfrac{S_n}{n}-\mu\right]\to 0$ almost surely.

I am able to find $\alpha$ for $p$ even and show convergence in $\mathbb{L}^p$ and almost sure. Any hints on how I can proceed with this problem? Thanks!

Ergodic
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  • We know right away, by CLT, that $\sup \alpha\leq 1/2$ for all $p$'s. – A.S. Oct 26 '15 at 08:59
  • Thanks! But the CLT holds for $p\ge 2$. What about $1\le p< 2$? I don't see how an $\alpha\le 1/2$ would work in that case. – Ergodic Oct 26 '15 at 12:47
  • Try searching for stable distributions, there is a deep theorem about that - http://tankonyvtar.ttk.bme.hu/pdf/46.pdf - there are a few words in this note – asomog Oct 26 '15 at 14:41
  • @Ergodic It's somewhat intuitive that by monotonicity $p_1<p_2, \implies \max a_{p_1}\le \max a_{p_2}$. – A.S. Oct 26 '15 at 16:36

1 Answers1

2

This problem can be solved by Marcinkiewicz-Zygmund SLLN, say

Suppose that $X_1,X_2,...$ i.i.d r.v.s and $1\leq p< 2$, and $S_n = \sum_{i=1}^nX_i$ defined as partial sums as usual, then

$$\frac{S_n-nc}{n^{1/p}}\to 0\quad \text{a.s. }\quad \text{for some constant $c$}$$

if and only if $\mathbb{E}|X_1|^p < \infty$ and $\mathbb{E}(X_1) =c$.

Thus if $1\leq p<2$, the largest $\alpha = 1 - 1/p$.

While if $p\geq 2$, by central limit theorem, $\sqrt{n}(\frac{S_n}{n}-\mu)\to N(0,\sigma^2)$, note that $\sigma^2$ exist by the fact that $\mathbb{E}|X_1|^p < \infty, p\geq 2$, so there does not exist the largest $\alpha$, the largest $\alpha$ is infinitely close to 1/2 but strictly less than 1/2.

Schrödinger's Cat
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