[Paper Review] Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations

Zhang Li|arXiv (Cornell University)|Aug 2, 2016

Probability and Risk Models27 references20 citations

TL;DR

This paper establishes strong limit theorems—specifically the strong law of large numbers and the law of the iterated logarithm—for sequences of extended independent and extended negatively dependent random variables under sub-linear expectations. It introduces weaker forms of dependence that are easier to verify than Peng’s original independence, derives new moment and exponential inequalities, and proves that the Choquet integral condition $ C_{\mathbb{V}}(|X_1|) < \infty $ is both necessary and sufficient for the strong law of large numbers under extended independence.

ABSTRACT

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have raised progressive interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng (2008b). We introduce a concept of extended negative dependence which is an extension of this kind of weak independence and the extended negative independence relative to classical probability appeared in recent literatures. Powerful tools as the moment inequality and Kolmogorov's exponential inequality are established for this kind of extended negatively independent random variables, which improve those of Chen, Chen and Ng(2010) a lot. And the strong law of large numbers and the law of iterated logarithm are obtained by applying these inequalities.

Motivation & Objective

• To extend the framework of limit theorems under sub-linear expectations beyond classical independence.
• To introduce and analyze a weaker form of stochastic independence—extended independence—that is easier to verify than Peng’s definition.
• To develop new moment and exponential inequalities tailored for extended negatively dependent random variables.
• To establish the strong law of large numbers and the law of the iterated logarithm under this extended dependence structure.
• To show that the Choquet integral condition $ C_{\mathbb{V}}(|X_1|) < \infty $ is both necessary and sufficient for the strong law of large numbers under extended independence.

Proposed method

• Introduces extended independence via the condition $ \widehat{\mathbb{E}}[\prod_{i=1}^n \psi_i(X_i)] = \prod_{i=1}^n \widehat{\mathbb{E}}[\psi_i(X_i)] $ for non-negative $ \psi_i \in C_{l,Lip}(\mathbb{R}) $.
• Defines extended negative dependence as a generalization of classical negative dependence in the context of sub-linear expectations.
• Establishes new moment inequalities and Kolmogorov-type exponential inequalities for extended negatively dependent random variables.
• Applies the Borel-Cantelli lemma and countable sub-additivity of the upper capacity $ \mathbb{V}^* $ to control tail probabilities of partial sums.
• Uses truncation techniques, setting $ Y_k = (-b_k) \vee (X_k \wedge b_k) $ with $ b_k = k^\beta $, to control large deviations.
• Employs a subsequence argument with $ n_k = [e^{k^{1-\alpha}}] $ to prove almost sure convergence via exponential bounds on partial sum increments.

Experimental results

Research questions

• RQ1 What is the weakest form of stochastic independence that still allows for strong limit theorems under sub-linear expectations?
• RQ2 Can the strong law of large numbers be established under extended independence with minimal moment conditions?
• RQ3 How can moment and exponential inequalities be generalized for extended negatively dependent random variables in non-linear expectation spaces?
• RQ4 Is the condition $ C_{\mathbb{V}}(|X_1|) < \infty $ both necessary and sufficient for the strong law of large numbers under extended independence?
• RQ5 Can the law of the iterated logarithm be proven for extended negatively dependent sequences under sub-linear expectations?

Key findings

• The strong law of large numbers holds under extended independence if and only if $ C_{\mathbb{V}}(|X_1|) < \infty $, extending Zhang (2016) to a weaker dependence structure.
• A new moment inequality and Kolmogorov-type exponential inequality are established for extended negatively dependent random variables, improving upon Chen, Chen, and Ng (2010).
• The law of the iterated logarithm is proven for extended negatively dependent sequences under sub-linear expectations, with the normalization $ a_n = \sqrt{2n \log \log n} $.
• The proof of the necessary condition for the strong law relies on a refined analysis of the tail capacity of partial sums and the use of a subsequence $ n_k = [e^{k^{1-\alpha}}] $.
• The convergence $ \mathbb{V}^*\left(\limsup_{n\to\infty} \frac{\sum_{k=1}^n (Y_k - \widehat{\mathbb{E}}[Y_k])}{a_n} > (1+\epsilon)^2\right) = 0 $ is established via exponential bounds and Borel-Cantelli arguments.
• The result holds under the assumption $ \widehat{\mathbb{E}}[X_1] = 0 $, $ \widehat{\mathbb{E}}[X_1^2] = 1 $, and $ \mathbb{E}[|X_1|^{2+\gamma}] < \infty $ for some $ \gamma > 0 $.