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Markov Processes Generated by Iterations of I.I.D. Maps

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Stationary Processes and Discrete Parameter Markov Processes

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 293))

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Abstract

While all discrete parameter Markov processes on a Polish state space can be represented as i.i.d. iterations of random maps, the properties of the maps obviously play a significant role in their long-run behavior. Non-decreasing monotonicity is one such property for which definitive results can be obtained, as illustrated in this chapter.

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Notes

  1. 1.

    See the specific subject matter texts by Ramasubramanian (2009) and by Rolski et al. (2010) for comprehensive treatments of the ruin problem and much more.

  2. 2.

    BCPT, p. 242.

  3. 3.

    BCPT, p. 34.

  4. 4.

    BCPT p.242, p. 34.

  5. 5.

    BCPT, Proposition 7.14, pp. 146,147.

  6. 6.

    BCPT, Lemma 4, p. 244.

  7. 7.

    See Bhattacharya and Ranga Rao (2010), p. 17.

  8. 8.

    BCPT, pp. 244–245, or Folland (1984), p. 131.

References

  • Bhattacharya R, Waymire E (2021) Random walk, Brownian motion, and martingales. Graduate text in mathematics. Springer, New York

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  • Dubins LE, Freedman DA (1966) Invariant probabilities for certain Markov processes. Ann Math Statist 37(4):837–848

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  • Ramasubramanian S (2009) Lectures on insurance models. American Mathematical Society, Providence

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  • Rolski T, Schmidli H, Schimidt V, Teugels JL (2010) Stochastic processes for insurance & finance. Wiley, New York

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Rabi Bhattacharya

  2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

    Edward Waymire

Authors
  1. Rabi Bhattacharya
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  2. Edward Waymire
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Bhattacharya, R., Waymire, E. (2022). Markov Processes Generated by Iterations of I.I.D. Maps. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_18

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