Geodesic
On renewal matrices connected with branching processes with tails of distributions of different orders
Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 139-192.

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We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property. We assume that the heaviest tails are regularly varying at the infinity with parameter $-\beta\in[-1, 0)$ and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices. 
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V. A. Topchiǐ. On renewal matrices connected with branching processes with tails of distributions of different orders. Matematičeskie trudy, Tome 20 (2017) no. 2, pp. 139-192. http://geodesic.mathdoc.fr/item/MT_2017_20_2_a6/

[1] Borovkov A. A., Teoriya veroyatnostei, 3-e izd., Editorial URSS, M., Novosibirsk, 1999

[2] Vatutin V. A., “Diskretnye predelnye raspredeleniya chisla chastits v vetvyaschikhsya protsessakh Bellmana–Kharrisa s neskolkimi tipami chastits”, TVP, 24:3 (1979), 503–514 | Zbl

[3] Vatutin V. A., “Ob odnom klasse kriticheskikh vetvyaschikhsya protsessov Bellmana–Kharrisa s neskolkimi tipami chastits”, TVP, 25:4 (1980), 771–781 | Zbl

[4] Vatutin V. A., Topchii V. A., “Kataliticheskie vetvyaschiesya sluchainye bluzhdaniya na $\mathbb{Z}^d$ s vetvleniem v nule”, Matem. tr., 14:2 (2011), 28–72 | Zbl

[5] Vatutin V. A., Topchii V. A., “Kriticheskie vetvyaschiesya protsessy Bellmana–Kharrisa s dolgo zhivuschimi chastitsami”, Tr. Matem. in-ta im. V. A. Steklova, 282, no. 2, 2013, 257–287 | Zbl

[6] Vatutin V. A., Topchii V. A., “Osnovnaya teorema vosstanovleniya dlya raspredelenii s tyazhelymi khvostami, imeyuschimi indeks $\beta\in(0,0.5]$”, TVP, 58:2 (2013), 387–396 | DOI

[7] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967

[8] Ilin V. A., Poznyak E. G., Lineinaya algebra, Nauka, M., 1999

[9] Rogozin B. A., Sgibnev M. S., “Banakhovy algebry absolyutno nepreryvnykh mer na pryamoi”, Sib. matem. zhurn. 1979, 20:1, 119–127

[10] Sgibnev M. S., “Banakhovy algebry mer klassa $\mathcal{G}(\gamma)$”, Sib. matem. zhurn., 29:4 (1988), 162–171

[11] Sevastyanov B. A., Vetvyaschiesya protsessy, Nauka, M., 1971

[12] Topchii V. A., “Proizvodnaya plotnosti vosstanovleniya s beskonechnym momentom pri $\alpha\in(0,1/2]$”, Sib. elektron. matem. izv., 7 (2010), 340–349

[13] Topchii V. A., “Asimptotika proizvodnykh ot funktsii vosstanovleniya dlya raspredelenii bez pervogo momenta s pravilno menyayuschimisya khvostami stepeni $\beta \in(0.5,1]$”, Diskret. matem., 24:2 (2012), 123–148 | DOI | Zbl

[14] Topchii V. A., “Teoremy dvumernogo vosstanovleniya pri slabykh momentnykh ogranicheniyakh i kriticheskie vetvyaschiesya protsessy Bellmana–Kharrisa”, Diskret. matem., 27:1 (2015), 123–145 | DOI

[15] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984

[16] Shurenkov V. M., “Zamechanie ob uravnenii mnogomernogo vosstanovleniya”, TVP, 20:4 (1975), 848–851 | Zbl

[17] Athreya K. B., Ney P. E., Branching Processes, Die Grundlehren der Mathematischen Wissenschaften, 196, Springer-Verlag, New York–Heidelberg, 1972 | MR | Zbl

[18] Bingham N. H., Goldie C. M., Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[19] Erickson K. B., “Strong renewal theorems with infinite mean”, Trans. Amer. Math. Soc., 151 (1970), 263–291 | DOI | MR | Zbl

[20] Topchii V., “Renewal measure density for distributions with regularly varying tails of order $\alpha\in(0,1/2]$”, Workshop on Branching Processes and Their Applications, Lecture Notes in Statistics, 197, Springer, Berlin, 2010, 109–118 | DOI | MR

[21] Vatutin V., Iksanov A., Topchii V., “A two-type Bellman–Harris process initiated by a large number of particles”, Acta Appl. Math., 138:1 (2015), 279–312 | DOI | MR | Zbl