Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations
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| Format: | Preprint |
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2024
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| _version_ | 1866913473555857408 |
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| author | Xu, Mingzhou Wang, Wei |
| author_facet | Xu, Mingzhou Wang, Wei |
| contents | Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. Assume that $h(\cdot)$ is a positive non-decreasing function on $(0,\infty)$ fulfulling $\int_{1}^{\infty}(th(t))^{-1}\dif t=\infty$. Write $Lt=\ln \max\{\me,t\}$, $ψ(t)=\int_{1}^{t}(sh(s))^{-1}\dif s$, $t\ge 1$. In this sequel, we establish that $\sum_{n=1}^{\infty}(nh(n))^{-1}\vv\left\{|S_n|\ge (1+\varepsilon)σ\sqrt{2nLψ(n)}\right\}<\infty$, $\forall \varepsilon>0$ if $\ee(X)=\ee(-X)=0$ and $\ee(X^2)=σ^2\in (0,\infty)$. The result generalizes that of NA random variables in probability space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10662 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations Xu, Mingzhou Wang, Wei Probability 60F15, 60G50 Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. Assume that $h(\cdot)$ is a positive non-decreasing function on $(0,\infty)$ fulfulling $\int_{1}^{\infty}(th(t))^{-1}\dif t=\infty$. Write $Lt=\ln \max\{\me,t\}$, $ψ(t)=\int_{1}^{t}(sh(s))^{-1}\dif s$, $t\ge 1$. In this sequel, we establish that $\sum_{n=1}^{\infty}(nh(n))^{-1}\vv\left\{|S_n|\ge (1+\varepsilon)σ\sqrt{2nLψ(n)}\right\}<\infty$, $\forall \varepsilon>0$ if $\ee(X)=\ee(-X)=0$ and $\ee(X^2)=σ^2\in (0,\infty)$. The result generalizes that of NA random variables in probability space. |
| title | Convergence rate in the law of logarithm for negatively dependent random variables under sub-linear expectations |
| topic | Probability 60F15, 60G50 |
| url | https://arxiv.org/abs/2408.10662 |