Constants in the Rosenthal inequality

Question

Let $X_1,\ldots,X_n$ be independent with $\mathbf{E}[X_i] = 0$ and $\mathbf{E}[|X_i|^t] < \infty$ for some $t \ge 2$. Write $X = \sum_{i=1}^n X_i$. Then we have the family of "Rosenthal-type inequalities":

$$ \mathbf{E}[|X|^t] \le C_1(t)\cdot \left(\sum_{i=1}^n \mathbf{E}[|X_i|^t]\right) + C_2(t)\cdot \left(\sum_{i=1}^n \mathbf{E}[X_i^2]\right)^{t/2} .$$

Henceforth, $c>0$ is some absolute constant. I have seen in various papers that we can, for example, take $C_1(t) = C_2(t) = (ct/\log(t))^t$. We can also take $C_1(t) = (ct)^t, C_2(t) = (c\sqrt{t})^t$. Another option is $C_1(t) = c^t, C_2(t) = c^t\cdot 2^{t^2/4}$.

Is it known whether there is some non-trivial tradeoff curve for the relationship between $C_1(t)$ and $C_2(t)$? For example, if I'm fine with setting $C_2(t) = (ct^{2/3})^t$, what's the best $C_1(t)$ I can get?

Answer 1

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\R}{\mathbb{R}} \newcommand{\X}{\mathcal{X}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$

According to Theorems 6.1 and 6.2, the best constants $C_1(t)$ and $C_2(t)$ are given by \begin{equation} C_1(t)=(c\ga)^t,\quad C_2(t)=(c\sqrt\ga\,e^{t/\ga})^t \end{equation} for $\ga\in[1,t]$, up to a universal real constant $c>0$.

E.g., if you want $C_2(t) = (ct^{2/3})^t$, solving the equation $(ct^{2/3})^t=(c\sqrt\ga\,e^{t/\ga})^t$, you get $\ga\sim\dfrac{6t}{\ln t}$ as $t\to\infty$, so that you get $C_1(t)=\Big(\dfrac{ct}{\ln t}\Big)^t$ for some (possibly different) universal real constant $c>0$.

Exact versions of Rosenthal-type bounds for $t\in[2,3]\cup[5,\infty)$ are given in Theorems 1.3 and 1.5; see also Proposition 1.2.

Answer 2

There is a good constant in Rio's Marcinkiewicz-Zygmund type inequality: $${\bf E} |X|^t \le \left((t-1)\sum_{i=1}^n ({\bf E}|X_i|^t)^{2/t} \right) ^{t/2} $$ for independent centered $X_i$, $t\ge 2$ . [ Rio, E., 2009, Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions, J. Theor. Probab. 22: 146-163. https://doi.org/10.1007/s10959-008-0155-9 ]