On the absolute continuity of the limit random variable in the supercritical Galton-Watson branching process
HTML articles powered by AMS MathViewer
- by Krishna B. Athreya PDF
- Proc. Amer. Math. Soc. 30 (1971), 563-565 Request permission
Abstract:
Let $\{ {Z_n}:n \geqq 0\}$ be a simple Galton-Watson branching process with offspring distribution $\{ {p_j}\}$ satisying $1 < \sum {j{p_j} < \infty }$. It is known that there exist constants ${C_n}$ such that ${W_n} \equiv {Z_n}{C_n}$ converges with probability one to a nondegenerate limit random variable W. Here we show that this W is always absolutely continuous on $(0,\infty )$.References
- Krishna B. Athreya and Peter E. Ney, Branching processes, Die Grundlehren der mathematischen Wissenschaften, Band 196, Springer-Verlag, New York-Heidelberg, 1972. MR 0373040
- Krishna B. Athreya and Peter Ney, The local limit theorem and some related aspects of super-critical branching processes, Trans. Amer. Math. Soc. 152 (1970), 233–251. MR 268971, DOI 10.1090/S0002-9947-1970-0268971-X
- C. C. Heyde, Extension of a result of Seneta for the super-critical Galton-Watson process, Ann. Math. Statist. 41 (1970), 739–742. MR 254929, DOI 10.1214/aoms/1177697127
- H. Kesten and B. P. Stigum, A limit theorem for multidimensional Galton-Watson processes, Ann. Math. Statist. 37 (1966), 1211–1223. MR 198552, DOI 10.1214/aoms/1177699266
- E. Seneta, On recent theorems concerning the supercritical Galton-Watson process, Ann. Math. Statist. 39 (1968), 2098–2102. MR 234530, DOI 10.1214/aoms/1177698037
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 563-565
- MSC: Primary 60.67
- DOI: https://doi.org/10.1090/S0002-9939-1971-0282421-5
- MathSciNet review: 0282421