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On Geometric-type Approximations with Applications

Author

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  • Fraser Daly

    (Heriot–Watt University)

  • Claude Lefèvre

    (Université Libre de Bruxelles)

Abstract

We explore two aspects of geometric approximation via a coupling approach to Stein’s method. Firstly, we refine precision and increase scope for applications by convoluting the approximating geometric distribution with a simple translation selected based on the problem at hand. Secondly, we give applications to several stochastic processes, including the approximation of Poisson processes with random time horizons and Markov chain hitting times. Particular attention is given to geometric approximation of random sums, for which explicit bounds are established. These are applied to give simple approximations, including error bounds, for the infinite-horizon ruin probability in the compound binomial risk process.

Suggested Citation

  • Fraser Daly & Claude Lefèvre, 2025. "On Geometric-type Approximations with Applications," Methodology and Computing in Applied Probability, Springer, vol. 27(1), pages 1-16, March.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:1:d:10.1007_s11009-024-10130-w
    DOI: 10.1007/s11009-024-10130-w
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    References listed on IDEAS

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    1. Asmussen, Soren & Avram, Florin & Usabel, Miguel, 2002. "Erlangian Approximations for Finite-Horizon Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 267-281, November.
    2. Michel, R., 1987. "An Improved Error Bound for the Compound Poisson Approximation of a Nearly Homogeneous Portfolio," ASTIN Bulletin, Cambridge University Press, vol. 17(2), pages 165-169, November.
    3. Daly, Fraser, 2019. "On strong stationary times and approximation of Markov chain hitting times by geometric sums," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 74-80.
    4. Gerber, Hans U., 1988. "Mathematical Fun with the Compound Binomial Process," ASTIN Bulletin, Cambridge University Press, vol. 18(2), pages 161-168, November.
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