Let $X_1,X_2,X_3,...$ be iid non-negative random variables with $E[X_i]=\infty$. I am looking for references on the growth in $n$ of the empirical average under assumptions on $X_1.,..,X_n$.

A more specific question is the following:

- Under the moment assumption $E[X_i^p]=1$ for some $p\in (0,1)$, what are deterministic $a_{n,p}\to_{n\to\infty}\infty$ such that $\frac{1}{n}\sum_{i=1}^n X_i = O_P(a_{n,p})$?

For $p=1/2$ a loose bound is given by applying the strong law of large numbers to $\sqrt{X_i}$ and then using $\sum_i X_i \le (\sum_i \sqrt{X_i})^2$; this gives $a_{n,1/2}=n$. I suspect much clever bounds exist.

(Above, $W_n=O_P(a_n)$ for $a_n>0$ if and only if $\forall \epsilon>0, \exists K_\epsilon>0$ such that $P(|W_n|>K a_n)\le \epsilon$.)