Embrechts–Goldie’s Problem on the Class of Lattice Convolution Equivalent Distributions

We show that the class of lattice convolution equivalent distributions is not closed under convolution roots. We prove that the class of lattice convolution equivalent distributions is closed under convolution roots under the assumption of the exponentially asymptotic decreasing condition. This result is extended to the class S Δ of Δ -subexponential distributions. As a corollary, we show that the class S Δ is closed under convolution roots in the class L Δ . Moreover, we prove that the class of lattice convolution equivalent distributions is not closed under convolutions. Finally, we give a survey on the closure under convolution roots of the other distribution classes.