We show that the class of convolution equivalent distributions and the class of locally subexponential distributions are not closed under convolution roots. It gives a negative answer to the classical conjectures of Embrechts and Goldie. Moreover, we establish two sufficient conditions in order that the class of convolution equivalent distributions is closed under convolution roots.
"The Wiener condition and the conjectures of Embrechts and Goldie." Ann. Probab. 47 (3) 1221 - 1239, May 2019. https://doi.org/10.1214/17-AOP1195