A simple proof, with a little-known extension, of a density version of Karamata's Tauberian theorem is presented, and the result applied to limit distributions of the Galton-Watson process.
"A Tauberian Theorem of E. Landau and W. Feller." Ann. Probab. 1 (6) 1057 - 1058, December, 1973. https://doi.org/10.1214/aop/1176996811