The limit points of the strong law of large numbers under the sub-linear expectations
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929201835147264 |
|---|---|
| author | Zhang, Li-Xin |
| author_facet | Zhang, Li-Xin |
| contents | Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean $\overlineμ $ and lower mean $\underlineμ $. Then for any Borel-measurable function $φ(x_1,\ldots,x_d)$ on $\mathbb R^d$ or continuous function $φ(x_1,x_2,\ldots)$ on $\mathbb R^{\mathbb N}$, $\sum_{i=1}^n X_i/n$ converges to $\underlineμ\wedge φ(X_1,X_2,\ldots)\wedge \overlineμ$ with upper capacity $1$. The limits of $\sum_{i=1}^nX_i/n$ can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of $(X_1, X_2,\ldots)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The limit points of the strong law of large numbers under the sub-linear expectations Zhang, Li-Xin Probability 60F15, 60F05 Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean $\overlineμ $ and lower mean $\underlineμ $. Then for any Borel-measurable function $φ(x_1,\ldots,x_d)$ on $\mathbb R^d$ or continuous function $φ(x_1,x_2,\ldots)$ on $\mathbb R^{\mathbb N}$, $\sum_{i=1}^n X_i/n$ converges to $\underlineμ\wedge φ(X_1,X_2,\ldots)\wedge \overlineμ$ with upper capacity $1$. The limits of $\sum_{i=1}^nX_i/n$ can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of $(X_1, X_2,\ldots)$. |
| title | The limit points of the strong law of large numbers under the sub-linear expectations |
| topic | Probability 60F15, 60F05 |
| url | https://arxiv.org/abs/2311.11100 |