A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory*

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn=-μ+∑j=-∞φn-jɛj, where {ɛ,ɛn;-∞<n<+∞}{ɛ,ɛn;-∞<n<+∞} is a sequence of independent, identically distributed random variables with zero mean, μ > 0 is a constant and the coefficients {φi;-∞<i<∞}{φi;-∞<i<∞} satisfy 0<∑j=-∞∞|jφj|<∞ Under the conditions that the distribution function of |ɛ| has dominated variation and ɛ satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-ημ+∑j=-∞∞ɛjβnj)>x} is discussed. Then the result is applied to ultimate ruin probability.