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An extension of a logarithmic form of Cramér's ruin theorem to some FARIMA and related processes

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  • Barbe, Ph.
  • McCormick, W.P.

Abstract

Cramér's theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér's. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.

Suggested Citation

  • Barbe, Ph. & McCormick, W.P., 2010. "An extension of a logarithmic form of Cramér's ruin theorem to some FARIMA and related processes," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 801-828, June.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:6:p:801-828
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    References listed on IDEAS

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    1. Hüsler, J. & Piterbarg, V., 2004. "On the ruin probability for physical fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 315-332, October.
    2. Burton, Robert M. & Dehling, Herold, 1990. "Large deviations for some weakly dependent random processes," Statistics & Probability Letters, Elsevier, vol. 9(5), pages 397-401, May.
    3. Promislow, S. David, 1991. "The probability of ruin in a process with dependent increments," Insurance: Mathematics and Economics, Elsevier, vol. 10(2), pages 99-107, July.
    4. Nyrhinen, Harri, 1995. "On the typical level crossing time and path," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 121-137, July.
    5. Muller, Alfred & Pflug, Georg, 2001. "Asymptotic ruin probabilities for risk processes with dependent increments," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 381-392, June.
    6. Gerber, Hans U., 1982. "Ruin theory in the linear model," Insurance: Mathematics and Economics, Elsevier, vol. 1(3), pages 213-217, July.
    7. Philippe Barbe & Michel Broniatowski, 1998. "Note on Functional Large Deviation Principle for Fractional ARIMA Processes," Statistical Inference for Stochastic Processes, Springer, vol. 1(1), pages 17-27, January.
    8. Collamore, Jeffrey F., 1998. "First passage times of general sequences of random vectors: A large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 78(1), pages 97-130, October.
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