The heavy-tail behavior of the difference of two dependent random variables

Abstract

Consider $Z=X-Y$, the difference of two nonnegative dependent random variables. We investigate how the difference $Z$ inherits the heavy tail property of the minuend $X$ and is altered by the subtrahend $Y$. In the case where $X$ and $Y$ are tail independent, we prove that if $X$ has a long tail ${\overline{F}}_X=1-F_X$, the asymptotic behavior of ${\overline{F}}_X$ is exactly inherited by $Z$, that is, ${\overline{F}}_Z\sim {\overline{F}}_X$, regardless of the tail behavior of $Y$. However, this result may not hold when $X$ and $Y$ exhibit tail dependence. Within the framework of bivariate regular variation, we show that the limit of the ratio $\frac{{\overline{F}}_Z}{{\overline{F}}_X}$ can range over the closed interval $[0,1]$.