Reduced-load equivalence for Gaussian processes (Q2488202)

Let \((X( t)) _{t\geq 0\text{ }}\)and \((Y( t) )_{t\geq 0} \) be independent Gaussian processes. Motivated by applications in queueing theory and insurance, it is important to derive the behaviour of \( P\left(\sup_{t\geq 0}[X(t)+Y(t) -ct]>u\right)\) as \(u\rightarrow \infty \) for a given \(c>0.\) The authors' main result provides necessary and sufficient conditions for to have \[ P\left(\sup_{t\geq 0}[X(t)+Y( t) -ct]>u\right)\sim P\left(\sup_{t\geq 0}[Y( t) -ct]>u\right) \] as \(u\rightarrow \infty \). If \(X\) and \(Y\) are fractional Brownian motions with Hurst parameters \(H_{X}\) and \(H_{Y}\), respectively, this result implies that the relation above holds when \(H_{Y}> {\frac12} ( H_{X}+1) \), and that it does not hold when \(H_{Y}\leq {\frac12} ( H_{X}+1) .\)