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Strong law of large numbers for $\varphi$-sub-Gaussian random variables under sub-linear expectation spaces

Nyanga Honda Masasila, István Fazekas

Abstract

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the definition of the sub-Gaussian property into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper framework for extending Zajkowski's general result to sublinear expectation spaces. Within our framework, we establish a strong law of large numbers for sub-Gaussian sequences. We present an example showing the usefulness of our results.

Strong law of large numbers for $\varphi$-sub-Gaussian random variables under sub-linear expectation spaces

Abstract

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the definition of the sub-Gaussian property into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper framework for extending Zajkowski's general result to sublinear expectation spaces. Within our framework, we establish a strong law of large numbers for sub-Gaussian sequences. We present an example showing the usefulness of our results.
Paper Structure (5 sections, 3 theorems, 32 equations)

This paper contains 5 sections, 3 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Basic Properties of Capacities and Sub-linear Expectations
  3. Sub-Gaussian Random Variables under Sub-linear Expectations
  4. The Main Result
  5. Strong law of large numbers for independent sub-Gaussian variables

Key Result

Lemma 3.3

Assume that upGauss and lowGauss are satisfied. Then for $\varepsilon>0$ we have

Theorems & Definitions (14)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Theorem 4.1
  • ...and 4 more