heavy-tailed concentration

In light-tailed settings, mean estimation is usually treated as synonymous with concentration, since the sample-mean is the natural estimate of the mean and we’re usually interested in concentration of the sum or mean of random variables. In heavy-tailed settings this is not the case, as the sample mean is a suboptimal estimate of the mean.

So when it comes to heavy-tailed distributions, we use estimates of the mean that are not the sample mean. These are summarized in scalar heavy-tailed mean estimation and multivariate heavy-tailed mean estimation.

There’s an interesting class of results for the sample mean under general distributional assumptions which rely on approximating arbitrary observations with truncated observations. Given $X_1,\dots ,X_n$, write

$$X_i=Y_i+Z_i,$$

where $Y_i=\min \{X_i,c\}$ and $Z_i=X_i-Y_i$. One can show that

$$\mathbb{P}({\overline{X}}_n\ge u)\le \mathbb{P}({\overline{Y}}_n\ge u)+\mathbb{P}(\underset{i}{\max }{X}_i\ge c).$$

Since the $Y_i$ are upper bounded, many well-known (one-sided) concentration inequalities are known for the first term on the right hand side; see bounded scalar concentration. So if one can control the second term on the rhs for particular values of $c$, this can lead to good concentration.

Inequalities of this type are known as Fuk-Nagaev inequalities. See eg Rio 2017 who explores some of them. There are also Banach space versions of the Fuk-Nagaev inequalities; see here and here. Hanh & Klass (1997, AoP) give similar style of results, giving tight and achievable upper bounds on the sum of iid random variables in terms of the conditional MGF.