An asymptotic property of the solution to the homogeneous generalized Wiener-Hopf equation (Q542197)

The following homogeneous generalized Wiener-Hopf equation is considered \[ S(x) = \int_{-\infty}^x {S(x - y)F(dy)},\quad x \geq 0. \tag{1} \] Here \(F\) is a probability distribution on \(\mathbb R\) with zero mean, finite variance, and infinite moment \(\int_{0}^{\infty } x^{3} F(dy)\). The author establishes that the \(P^*\)-solution \(S(x)\) of (1) enjoys the property \[ S(x) - ax \sim b\int_0^x \int_y^\infty \int_v^\infty F((u,\infty ))dudvdy \quad \text{as }x \to \infty, \] where \(a\) and \(b\) are explicit positive constants.