Strong law of large numbers for weighted sums of $ m $-widely acceptable random variables under sub-linear expectation space

Qingfeng Wu; Xili Tan; Shuang Guo; Peiyu Sun; Qingfeng Wu; Xili Tan; Shuang Guo; Peiyu Sun

doi:10.3934/math.20241442

AIMS Mathematics

2024, Volume 9, Issue 11: 29773-29805. doi: 10.3934/math.20241442

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Strong law of large numbers for weighted sums of $ m $-widely acceptable random variables under sub-linear expectation space

College of Mathematics and Statistics, Beihua University, Jilin 132013, China

Received: 03 August 2024 Revised: 11 October 2024 Accepted: 16 October 2024 Published: 21 October 2024
MSC : 60F15

In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $ m $-widely acceptable ($ m $-WA, for short) random variables under sublinear expectation space with the integral condition
$ \hat{\mathbb{E}} \left ( f^-\left ( \left | X \right | \right ) \right ) \le \mathrm{C}_\mathbb{V}\left ( f^-\left ( \left | X \right | \right ) \right )< \infty $
and $ Choquet $ integrals existence, respectively, where
$ f\left ( x \right ) = x^{1/\beta }L\left ( x \right ) $
for $ \beta > 1 $, $ L\left (x \right) > 0 $ $ \left(x > 0\right) $ was a monotonic nondecreasing slowly varying function, and $ f^-\left (x \right) $ was the inverse function of $ f\left(x\right) $. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $ m $-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $ m $-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.
- $ m $-widely acceptable random variables,
- strong law of large numbers,
- sublinear expectation space,
- slowly varying function,
- almost surely convergence
Citation: Qingfeng Wu, Xili Tan, Shuang Guo, Peiyu Sun. Strong law of large numbers for weighted sums of $ m $-widely acceptable random variables under sub-linear expectation space[J]. AIMS Mathematics, 2024, 9(11): 29773-29805. doi: 10.3934/math.20241442

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Abstract

In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $ m $-widely acceptable ($ m $-WA, for short) random variables under sublinear expectation space with the integral condition

$ \hat{\mathbb{E}} \left ( f^-\left ( \left | X \right | \right ) \right ) \le \mathrm{C}_\mathbb{V}\left ( f^-\left ( \left | X \right | \right ) \right )< \infty $

and $ Choquet $ integrals existence, respectively, where

$ f\left ( x \right ) = x^{1/\beta }L\left ( x \right ) $

for $ \beta > 1 $, $ L\left (x \right) > 0 $ $ \left(x > 0\right) $ was a monotonic nondecreasing slowly varying function, and $ f^-\left (x \right) $ was the inverse function of $ f\left(x\right) $. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $ m $-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $ m $-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.

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