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

TL;DR

This paper extends Cramer's ruin theorem to (g,F)-processes, including FARIMA models, providing tail probability estimates for their maximum and characterizing the most likely paths leading to large maxima.

Contribution

It generalizes Cramer's theorem to a broader class of processes like FARIMA, offering new tail estimates and path characterizations for these models.

Findings

Logarithmic tail probability estimates for (g,F)-processes with negative drift.

Identification of most likely paths leading to large maxima.

Extension of classical ruin results to fractional ARIMA models.

Abstract

Cramer'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 Cramer's. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.