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Cramer's theorem provides an estimate for the tail probability of the maximumof a random walk with negative drift and increments having a moment generatingfunction finite in a neighborhood of the origin. The class of (g,F)-processesgeneralizes in a natural way random walks and fractional ARIMA models used intime series analysis. For those (g,F)-processes with negative drift, we obtaina logarithmic estimate of the tail probability of their maximum, underconditions comparable to Cramer's. Furthermore, we exhibit the most likelypaths as well as the most likely behavior of the innovations leading to a largemaximum.

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