The law of large numbers and renewal process for T-related weighted fuzzy numbers on image

This paper considers the law of large numbers for T-related weighted fuzzy variables whose underlying spaces are image. We provide new sufficient conditions that do not depend on the shape of fuzzy numbers and the additive generator of a t-norm T for the law of large numbers for T-related weighted fuzzy variables. The results provide a far-reaching generalization of previous findings on the law of large numbers for T-related fuzzy numbers. In addition, we prove T-related fuzzy renewal theorems in which the inter-arrival time is characterized as weighted fuzzy numbers under t-norm-based fuzzy operations on image by using the law of large numbers for weighted fuzzy variables on image. We also investigate the law of large numbers and the renewal process for T-related fuzzy numbers on image with respect to the credibility measure, the chance measure, and the expected value.