# Concentration inequalities for heavy-tailed distributions

Suppose $$X_1,...,X_N$$ are $$N$$ i.i.d random variables with heavy tailed distributions. For example, $$E[X_i^p]\leq 1$$ for some $$p\geq 1$$. Are there some concentration inequalities to bound the tail

$$P(\sum_{i=1}^N X_i>t)$$

for $$t>0$$?

If $$X_i$$ are sub-Gaussian or sub-exponential, than standard concentration inequalities such as Chernoff bounds and Hoeffding's inequality work well. So how about only requiring $$E[X_i^p]$$ to be bounded?

The standard way to proceed here is by right-tail truncation (see e.g. Section 1 of Nagaev's paper), as follows.

Suppose that $$EX_i=0$$ for all $$i$$; the condition that the $$X_i$$'s are identically distributed will not be used in this answer. Let $$S_N:=\sum_{i=1}^N X_i$$. Let $$T_N:=\sum_{i=1}^N Y_i$$, where $$Y_i:=X_i\,1(X_i\le b)$$ for some real $$b>0$$. Then \begin{align} P(S_N\ge t)&\le P(\max_{i=1}^N X_i>b)+P(T_N\ge t) \\ &\le\sum_{i=1}^N P(X_i>b)+P(T_N\ge t). \tag{0}\label{0} \end{align}

Finally, to upper-bound $$P(T_N\ge t)$$ for real $$t\ge0$$, one can use e.g. Hoeffding's bound (2.9), to get \begin{align} P(T_N\ge t) &\le\exp\Big(-\frac tb\, g\Big(\frac{B^2}{bt}\Big)\Big) \tag{10}\label{10} \\ &\le\Big(\frac{B^2}{bt}\Big)^{ct/b}, \tag{20}\label{20} \end{align} where $$B:=\sqrt{\sum_{i=1}^N EX_i^2}$$, $$g(u):=(1+u)\ln(1+\frac1u)-1$$, and $$c:=\min_{0. In particular, choosing $$b=ct/p$$ for an arbitrary real $$p>0$$, from \eqref{20} we get $$P(T_N\ge t)\le\Big(\frac pc \frac{B^2}{t^2}\Big)^p. \tag{30}\label{30}$$

The bound in \eqref{10} is the best bound in its terms based on the Bernstein--Chernoff inequality. Other, similar, more general, and/or better results can be found e.g. in mentioned Section 1 of of Nagaev's paper and in this paper; see also references therein.

Another bound on $$P(S_N\ge t)$$ for real $$t>0$$ can be obtained from the Rosenthal inequality -- see e.g. this question and this answer: $$P(S_N\ge t)\le\frac{E|S_N|^p}{t^p} \le \frac{K(p)}{t^p}\,(A_p+B^p)$$ for real $$p\ge2$$, where $$K(p)$$ is a real number depending only on $$p$$ and $$A_p:=\sum_{i=1}^N E|X_i|^p$$. (Actually, the Rosenthal bound can be derived from \eqref{0} and \eqref{30}.)

For $$p\in[1,2]$$, one can similarly use the von Bahr--Esseen inequality, to get $$P(S_N\ge t)\le\frac{E|S_N|^p}{t^p} \le \frac{L(p)}{t^p}\,A_p,$$ where $$L(p)\in[1,2]$$ depends only on $$p$$.

• Thanks for this great solution. Commented Mar 14 at 8:27
• @jkfds : I am glad you liked the answer. In such a case, these guidelines may be relevant. Commented Mar 15 at 2:16