An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes |
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Authors: | Ph Barbe WP McCormick |
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Institution: | 1. CNRS (UMR8088), 90 rue de Vaugirard, 75006 Paris, France;2. University of Georgia, Department of Statistics, Athens, GA 30602, United States |
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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. |
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Keywords: | primary 60F99 secondary 60G70 60G50 62P05 60F10 60K25 60K99 |
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