Sharp Decoupling Inequalities for the Variances and Second Moments of Sums of Dependent Random Variables

TL;DR

This work develops sharp decoupling inequalities for the variances and second moments of sums of dependent random variables. It introduces a new proof of the complete decoupling lower bound for nonnegative sums and establishes sharp tangent decoupling upper bounds that do not require nonnegativity. The resulting inequalities bound the variance and second moment of dependent sums by those of their decoupled versions, enabling Chebyshev-type and Paley-Zygmund-type bounds, as well as bounds for randomly stopped sums. The findings generalize classical results to broader dependence structures and have practical implications for probabilistic bounds in statistics and stochastic analysis.

Abstract

Both complete decoupling and tangent decoupling are classical tools aiming to compare two random processes where one has a weaker dependence structure. We give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables $\sum\limits^n_{i=1} d_i$ \[ \frac{1}{2} \mathbb E \left( \sum\limits^n_{i=1} z_i \right)^2 \leq \mathbb E \left( \sum\limits^n_{i=1} d_i \right)^2, \] where $z_i \stackrel{\mathcal{L}}{=} d_i$ for all $i\leq n$ and $z_i$'s are mutually independent. We will then provide the following sharp tangent decoupling inequalities \[\mathbb Var \left( \sum\limits^n_{i=1} d_i\right) \leq 2 \mathbb Var \left( \sum\limits^n_{i=1} e_i\right),\] and \[\mathbb E \left( \sum\limits^n_{i=1} d_i\right)^2 \leq 2 \mathbb E \left( \sum\limits^n_{i=1} e_i\right)^2 - \left[ \mathbb E \left( \sum\limits^n_{i=1} e_i\right) \right]^2,\] where $\{e_i\}$ is the decoupled sequences of $\{d_i\}$ and $d_i$'s are not forced to be nonnegative. Applications to construct Chebyshev-type inequality and Paley-Zygmund-type inequality, and to bound the second moments of randomly stopped sums will be provided.

Theorems & Definitions (19)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• Theorem 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lem: two l2 norm']}
• Theorem 3.3: Decoupling inequality for the second moment