I am looking for references in the growth in $n$ of the empirical average $\bar X = \frac 1 n \sum_{i=1}^n X_i$ where $X_i$ are iid non-negative random variables with unbounded expectation.
For instance, if $E[\sqrt{X_i}] = a$ is finite, we may apply the law of large numbers to $\frac 1 n \sum_{i=1}^n \sqrt{X_i}$ and a very loose bound that $$\bar X \le \frac 1 n (\sum_{i=1}^n \sqrt{X_i} )^2 = n (a+o_P(1))^2.$$ Similarly if $E[X_i^p]=a$ for $p\in (0,1)$ then $$\bar X \le \frac 1 n (\sum_{i=1}^n X_i^p)(n\bar X)^{1-p}$$ so $(\bar X)^p \le n^{1-p} (a+o_p(1))$ and $\bar X = O_p(n^{1/p - 1})$.
It is hopefully possible to obtain more precise results than these loose bounds. What are references on the subject or techniques that would yield better bounds?