Geodesic
Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes
Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 47-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Fractional iteration of probability generating functions is investigated. In particular, conditions on the generating function of a Galton–Watson process that are necessary and sufficient for the process to admit imbedding in a continuous-time homogeneous Markov branching process are obtained. Necessary imbedding conditions formulated in terms of the several initial coefficients of the generating function are also obtained. The collection of all probability generating functions is partitioned, in accordance with a classification of branching processes, into subsets, and the latter are described as convex hulls of their extreme points. A description is given of the infinitesimal generators of distinguished semigroups of probability generating functions. 
@article{SM_1994_79_1_a3,
     author = {V. V. Goryainov},
     title = {Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes},
     journal = {Sbornik. Mathematics},
     pages = {47--61},
     year = {1994},
     volume = {79},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_79_1_a3/}
}
TY  - JOUR
AU  - V. V. Goryainov
TI  - Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes
JO  - Sbornik. Mathematics
PY  - 1994
SP  - 47
EP  - 61
VL  - 79
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1994_79_1_a3/
LA  - en
ID  - SM_1994_79_1_a3
ER  - 
%0 Journal Article
%A V. V. Goryainov
%T Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes
%J Sbornik. Mathematics
%D 1994
%P 47-61
%V 79
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1994_79_1_a3/
%G en
%F SM_1994_79_1_a3
V. V. Goryainov. Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes. Sbornik. Mathematics, Tome 79 (1994) no. 1, pp. 47-61. http://geodesic.mathdoc.fr/item/SM_1994_79_1_a3/

[1] Sevastyanov B. A., Vetvyaschiesya protsessy, Nauka, M., 1971 | MR

[2] Schroeder E., “Űber itierte Funktionen”, Math. Ann., 3 (1871), 296–322 | DOI

[3] Koenigs G., “Recherches sur les integrales des certaines equations fonctionelles”, Ann. Ecole Norm. Sup., 1(3) (1884), 3–41 | MR | Zbl

[4] Hadamard J., “Two works on iteration and related questions”, Bull. Amer. Math. Soc., 50 (1944), 67–75 | DOI | MR | Zbl

[5] Berkson E., Porta H., “Semigroups of analytic functions and composition operators”, Michigan Math. J., 25 (1978), 101–115 | DOI | MR | Zbl

[6] Cowen C. C., “Iteration and the solution of functional equations for functions analytic in the unit disk”, Trans. Amer. Math. Soc., 265 (1981), 69–95 | DOI | MR | Zbl

[7] Goryainov V. V., “Drobnye iteratsii analiticheskikh v edinichnom kruge funktsii s zadannymi nepodvizhnymi tochkami”, Matem. sb., 182 (1991), 1281–1299 | MR

[8] Kharris T., Teoriya vetvyaschikhsya sluchainykh protsessov, Mir, M., 1966

[9] Athreya K. B., Ney P. E., Branching processes, Springer-Verlag, Berlin, 1972 | MR | Zbl

[10] Karlin S., McGregor J., “Embeddability of discrete time simple branching processes into continuous time processes”, Trans. Amer. Math. Soc., 132 (1968), 115–136 | DOI | MR | Zbl

[11] Karlin S., McGregor J., “Embedding iterates of analytic functions with two fixed points into continuous groups”, Trans. Amer. Math. Soc., 132 (1968), 137–145 | DOI | MR | Zbl

[12] Goryainov V. V., “Polugruppy analiticheskikh funktsii i vetvyaschiesya protsessy”, DAN SSSR, 318 (1991), 1046–1049 | MR | Zbl

[13] Danford N., Shvarts Dzh., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[14] Valiron Zh., Analiticheskie funktsii, GITTL, M., 1957

[15] Ahlfors L. V., Complex analysis, McGraw-Hill, New York, 1966 | MR | Zbl