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Abstract
We study the limit behaviour of upper and lower bounds on expected time averages in imprecise Markov chains; a generalised type of Markov chain where the local dynamics, traditionally characterised by transition probabilities, are now represented by sets of ‘plausible’ transition probabilities. Our main result is a necessary and sufficient condition under which these upper and lower bounds, called upper and lower expected time averages, will converge as time progresses towards infinity to limit values that do not depend on the process’ initial state. Remarkably, our condition is considerably weaker than those needed to establish similar results for so-called limit—or steady state—upper and lower expectations, which are often used to provide approximate information about the limit behaviour of time averages as well. We show that such an approximation is sub-optimal and that it can be significantly improved by directly using upper and lower expected time averages.
Keywords
Imprecise Markov chain, Upper expectation, Upper transition operator, Expected time average, Weak Ergodicity

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MLA
T’Joens, Natan, and Jasper De Bock. “Limit Behaviour of Upper and Lower Expected Time Averages in Discrete-Time Imprecise Markov Chains.” Information Processing and Management of Uncertainty in Knowledge-Based Systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II, edited by Marie-Jeanne Lesot, vol. 1238, Springer, 2020, pp. 224–38, doi:10.1007/978-3-030-50143-3_17.
APA
T’Joens, N., & De Bock, J. (2020). Limit behaviour of upper and lower expected time averages in discrete-time imprecise Markov chains. In M.-J. Lesot (Ed.), Information processing and management of uncertainty in knowledge-based systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II (Vol. 1238, pp. 224–238). Springer. https://doi.org/10.1007/978-3-030-50143-3_17
Chicago author-date
T’Joens, Natan, and Jasper De Bock. 2020. “Limit Behaviour of Upper and Lower Expected Time Averages in Discrete-Time Imprecise Markov Chains.” In Information Processing and Management of Uncertainty in Knowledge-Based Systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II, edited by Marie-Jeanne Lesot, 1238:224–38. Springer. https://doi.org/10.1007/978-3-030-50143-3_17.
Chicago author-date (all authors)
T’Joens, Natan, and Jasper De Bock. 2020. “Limit Behaviour of Upper and Lower Expected Time Averages in Discrete-Time Imprecise Markov Chains.” In Information Processing and Management of Uncertainty in Knowledge-Based Systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II, ed by. Marie-Jeanne Lesot, 1238:224–238. Springer. doi:10.1007/978-3-030-50143-3_17.
Vancouver
1.
T’Joens N, De Bock J. Limit behaviour of upper and lower expected time averages in discrete-time imprecise Markov chains. In: Lesot M-J, editor. Information processing and management of uncertainty in knowledge-based systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II. Springer; 2020. p. 224–38.
IEEE
[1]
N. T’Joens and J. De Bock, “Limit behaviour of upper and lower expected time averages in discrete-time imprecise Markov chains,” in Information processing and management of uncertainty in knowledge-based systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II, 2020, vol. 1238, pp. 224–238.
@inproceedings{8666839,
  abstract     = {We study the limit behaviour of upper and lower bounds on expected time averages in imprecise Markov chains; a generalised type of Markov chain where the local dynamics, traditionally characterised by transition probabilities, are now represented by sets of ‘plausible’ transition probabilities. Our main result is a necessary and sufficient condition under which these upper and lower bounds, called upper and lower expected time averages, will converge as time progresses towards infinity to limit values that do not depend on the process’ initial state. Remarkably, our condition is considerably weaker than those needed to establish similar results for so-called limit—or steady state—upper and lower expectations, which are often used to provide approximate information about the limit behaviour of time averages as well. We show that such an approximation is sub-optimal and that it can be significantly improved by directly using upper and lower expected time averages.},
  author       = {T'Joens, Natan and De Bock, Jasper},
  booktitle    = {Information processing and management of uncertainty in knowledge-based systems, 18th International Conference, IPMU 2020, Lisbon, Portugal, June 15-19, 2020, Proceedings, Part II},
  editor       = {Lesot, Marie-Jeanne},
  isbn         = {9783030501525},
  issn         = {1865-0929},
  keywords     = {Imprecise Markov chain,Upper expectation,Upper transition operator,Expected time average,Weak Ergodicity},
  language     = {eng},
  pages        = {224--238},
  publisher    = {Springer},
  title        = {Limit behaviour of upper and lower expected time averages in discrete-time imprecise Markov chains},
  url          = {http://dx.doi.org/10.1007/978-3-030-50143-3_17},
  volume       = {1238},
  year         = {2020},
}

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