Abstract
Recently, Zhao et al. (in Fuzzy Optimization and Decision Making 2007 6, 279–295) presented a fuzzy random elementary renewal theorem and fuzzy random renewal reward theorem in the fuzzy random process. In this paper, we study the convergence of fuzzy random renewal variable and of the total rewards earned by time t with respect to the extended Hausdorff metrics d ∞ and d 1. Using this convergence information and applying the uniform convergence theorem, we provide some new versions of the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem.
Similar content being viewed by others
References
Colubi A., Lápez-Dáaz M., Dominguez-Menchero J. S., Gil M. A. (1999) A generalized strong law of large numbers. Probability Theory Related Fields 114: 401–417
Gao J., Liu B. (2001) New primitive chance measures of fuzzy random event. International Journal of Fuzzy systems 3: 527–531
Hong D. H. (2003) A convergence of fuzzy random variables. Kybernetika 39: 275–280
Hong D. H. (2006) Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards. Information Sciences 176: 2386–2395
Hong D. H. (2007) Renewal process for fuzzy variables. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15: 493–501
Hwang C. A. (2000) Theorem of renewal process for fuzzy random variables and its application. Fuzzy Sets and Systems 116: 237–244
Klement E. P., Puri L. M., Ralescu D. A. (1986) Limit theorems for fuzzy random variables. Proceedings of the Royal Society of London 407: 171–182
Li X., Liu B. (2006) New independence definition of fuzzy random variable and random fuzzy variable. World Journal of Modeling and Simulation, 2: 338–342
Liu B., Liu Y. (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10: 445–450
Liu Y., Liu B. (2003) Fuzzy random variables: A scalar expected value operator. Fuzzy Optimization and Decision Making 2: 143–160
Liu Y., Gao J. (2007) The dependence of fuzzy variables with applications to fuzzy random optimization. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 15(SUPPL): 1–20
Molchanov I. S. (1999) On strong law of large numbers for random upper semi-continuous functions. Journal of Mathematical Analysis and Applications 235: 349–355
Popova E., Wu H. (1999) Renewal reward processes with fuzzy rewards and their applications to T-age replacement policies. European Journal of Operational Research 117: 606–617
Ross S. (1996) Stochastic processes. John Wiley & Sons, New York
Zadeh L. A. (1978) Fuzzy Sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28
Zhao R., Liu B. (2003) Renewal process with fuzzy interarrival times and rewards. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 11: 573–586
Zhao R., Tang W. (2006) Some properties of fuzzy random processes. IEEE Transactions on Fuzzy Systems 2: 173–179
Zhao R., Tang W., Wang C. (2007) Fuzzy random renewal process and renewal reward process. Fuzzy Optimization and Decision Making 6: 279–295
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hong, D.H. Uniform convergence of fuzzy random renewal process. Fuzzy Optim Decis Making 9, 275–288 (2010). https://doi.org/10.1007/s10700-010-9085-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-010-9085-y