no. 1
On the limiting distribution of non negative additive functions
Compositio Mathematica, Tome 43 (1981) no. 1, pp. 101-105. 
@article{CM_1981__43_1_101_0,
     author = {Daboussi, H\'edi},
     title = {On the limiting distribution of non negative additive functions},
     journal = {Compositio Mathematica},
     pages = {101--105},
     publisher = {Sijthoff et Noordhoff International Publishers},
     volume = {43},
     number = {1},
     year = {1981},
     zbl = {0458.10037},
     mrnumber = {631429},
     language = {en},
     url = {archive.numdam.org/item/CM_1981__43_1_101_0/}
}
Daboussi, H. On the limiting distribution of non negative additive functions. Compositio Mathematica, Tome 43 (1981) no. 1, pp. 101-105. http://archive.numdam.org/item/CM_1981__43_1_101_0/

[1] P.D.T.A. Elliott: On the limiting distribution of f (p + 1) for non negative additive functions. Acta Arithmetica XXV (1974) 259-264. | MR 347760 | Zbl 0278.10050

[2] P. Erdös and A. Wintner: Additive arithmetical functions and statistical independence. Amer. J. Math. 61 (1939) 713-721. | JFM 65.0165.01 | MR 247 | Zbl 0022.00903

[3] I. Kátai: On the distribution of arithmetical functions on the set of primes plus one. Compositio Math. 19 (1968) 278-289. | Numdam | MR 244181 | Zbl 0167.04301