[Paper Review] Strong limit theorems for weighted sums of negatively associated random variables in nonlinear probability

Yuting Lan, Ning Zhang|arXiv (Cornell University)|Jun 19, 2017

Probability and Risk Models14 references3 citations

TL;DR

This paper establishes strong limit theorems for weighted sums of negatively associated random variables in nonlinear probability spaces without requiring independence or identical distribution. It derives Marcinkiewicz-Zygmund and Kolmogorov-type strong laws of large numbers under upper-lower expectation frameworks, and applies these to prove Strassen-type invariance principles for negatively associated and vertically independent random variables.

ABSTRACT

In this paper, based on the initiation of the notion of negatively associated random variables under nonlinear probability, a strong limit theorem for weighted sums of random variables within the same frame is achieved without assumptions of independence and identical distribution, from which the Marcinkiewich-Zygmund type and Kolmogorov type strong laws of large numbers are derived. In addition, as applications of our results, Stranssen type invariance principles of negatively associated random variables and vertically independent random variables are proposed respectively.

Motivation & Objective

• To extend classical strong laws of large numbers (SLLNs) to nonlinear probability spaces where additivity of expectation does not hold.
• To remove the assumptions of independence and identical distribution in SLLNs for negatively associated random variables under nonlinear expectations.
• To derive Marcinkiewicz-Zygmund and Kolmogorov-type SLLNs in the context of upper-lower expectations.
• To establish Strassen-type invariance principles for negatively associated and vertically independent random variables using the new SLLN results.
• To generalize existing SLLNs in nonlinear probability by using moment conditions and the concept of negative association under non-additive measures.

Proposed method

• Introduces a framework of nonlinear probability using upper-lower expectations derived from a family of probability measures.
• Defines negatively associated random variables in the nonlinear probability setting, extending classical dependence concepts to non-additive frameworks.
• Applies moment conditions: $\sup_{i\geq 1}\mathbb{E}[|X_i|^{\alpha+1}] < \infty$ for some $\alpha > 0$, to control tail behavior.
• Uses the structure of upper and lower expectations $\mathbb{E}[X] = \sup_{P\in\mathcal{P}} E_P[X]$, $\mathcal{E}[X] = \inf_{P\in\mathcal{P}} E_P[X]$ to derive pathwise almost sure convergence.
• Applies limit theorems to weighted sums $\sum_{i=1}^n a_i(X_i - \mathbb{E}[X_i])$ and $\sum_{i=1}^n a_i(X_i - \mathcal{E}[X_i])$ with normalization $A_n$.
• Proves invariance principles by analyzing the limsup and liminf of functionals of normalized weighted sums under lower probability $v$.

Experimental results

Research questions

• RQ1 Can strong laws of large numbers be established for weighted sums of negatively associated random variables in nonlinear probability without assuming independence or identical distribution?
• RQ2 How do Marcinkiewicz-Zygmund and Kolmogorov-type SLLNs extend under upper-lower expectations in nonlinear probability?
• RQ3 What conditions on moments and normalization sequences ensure almost sure convergence of weighted sums in nonlinear frameworks?
• RQ4 Can Strassen-type invariance principles be derived from the new SLLN results for dependent random variables in nonlinear probability?
• RQ5 How do the results apply to vertically independent random variables under nonlinear expectations?

Key findings

• The paper establishes a strong law of large numbers for weighted sums of negatively associated random variables under nonlinear probability, showing that $\mathbb{V}\left(\left\{\liminf_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathcal{E}[X_i])}{n} < 0\right\} \cup \left\{\limsup_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathbb{E}[X_i])}{n} > 0\right\}\right) = 0$.
• For $1 \leq p < 1 + \alpha$, the Marcinkiewicz-Zygmund SLLN holds: $\mathbb{V}\left(\left\{\liminf_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathcal{E}[X_i])}{n^{1/p}} < 0\right\} \cup \left\{\limsup_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathbb{E}[X_i])}{n^{1/p}} > 0\right\}\right) = 0$.
• The lower probability satisfies $v\left(\liminf_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathcal{E}[X_i])}{n^{1/p}} \geq 0\right) = 1$ and $v\left(\limsup_{n\to\infty}\frac{\sum_{i=1}^n(X_i - \mathbb{E}[X_i])}{n^{1/p}} \leq 0\right) = 1$.
• A Strassen-type invariance principle is proven: for continuous $\varphi$, $v\left(\limsup_{n\to\infty}\varphi\left(\frac{\sum_{i=1}^n a_i(X_i - \mathbb{E}[X_i])}{A_n}\right) \leq \sup_{x\leq 0}\varphi(x)\right) = 1$.
• The same invariance principle holds for vertically independent random variables under the same moment and normalization conditions.
• The results are derived under the condition $\lim_{n\to\infty}\frac{A_n}{n^{1/(\beta+1)}} = \infty$ with $\beta \in (0, \min(1,\alpha))$, ensuring sufficient growth of normalization.