Local lower large deviations of strongly supercritical BPREG

K. Yu. Denisov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 1-16

Résumé

We consider local probabilities of lower deviations for branching process $Z_{n} = X_{n, 1} + \dotsb + X_{n, Z_{n-1}}$ in random environment $\boldsymbol\eta$. We assume that $\boldsymbol\eta$ is a sequence of independent identically distributed variables and for fixed $\boldsymbol\eta$ the distribution of variables $X_{i,j}$ is geometric. We suppose that the associated random walk $S_n = \xi_1 + \dotsb + \xi_n$ has positive mean $\mu$ and satisfies left-hand Cramer's condition ${\mathbf E}\exp(h\xi_i) < \infty$ as $h^{-}<h<0$ for some $h^{-} < -1$. Under these assumptions, we find the asymptotic representation for local probabilities ${\mathbf P}\left( Z_n = \lfloor\exp\left(\theta n\right)\rfloor \right)$, where $\theta$ is near the boundary of the first and the second deviations zones.

Détail

Keywords: branching processes, random environment, random walk, Cramer's condition, large deviations, local theorems.

Bibliographie

[1] M.V. Kozlov, “On large deviations of branching processes in a random environment: geometric distribution of descendants”, Discrete Math. Appl., 16:2 (2006), 155–174 | DOI | MR | Zbl

[2] M.V. Kozlov, “On large deviations of strictly subcritical branching processes in a random environment with geometric distribution of progeny”, Theory Probab. Appl., 54:3 (2010), 424–446 | DOI | MR | Zbl

[3] V. Bansaye, J. Berestycki, “Large deviations for branching processes in random environment”, Markov Process. Relat. Fields, 15:4 (2009), 493–524 http://math-mprf.org/journal/articles/id1192/ | MR | Zbl

[4] D. Buraczewski, P. Dyszewski, Precise large deviation estimates for branching process in random environment, 2017, arXiv: 1706.03874 | MR

[5] A.V. Shklyaev, “Large deviations of branching process in a random environment. II”, Diskrete Math. Appl., 31:6 (2021), 431–447 | DOI | MR | Zbl

[6] V. Bansaye, C. Böinghoff, “Lower large deviations for supercritical branching processes in random environment”, Proc. Steklov Inst. Math., 282:1 (2013), 15–34 | DOI | MR | Zbl

[7] A.A. Borovkov, Asymptotic analysis of random walks. Rapidly decreasing distributions of increments, Fizmatlit, M., 2013 https://www.rfbr.ru/rffi/ru/books/o_1896836#1 | Zbl

[8] A. Agresti, “On the extinction times of varying and random environment branching processes”, J. Appl. Probab., 12:1 (1975), 39–46 https://www.jstor.org/stable/3212405?casa_token=kLGwYB3ozfkAAAAA:w31MN1gZPHoMqec_nqO93y48EtdREYiz42vDY4FH1vopSNwQEh9D6rPHeŻW0ISGLWRotQQJ5Yzc2mDFreihAtSHy-7ourXANOxqis11o1E699efca44 | DOI | MR | Zbl

[9] K.Yu. Denisov, “Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants”, Diskrete Math. Appl., 32:5 (2022), 313–323 | DOI | MR | Zbl

[10] K.Yu. Denisov, “Local lower deviations of strictly supercritical branching process in random environment with geometric number of descendants”, Diskretnaya Matematika, 34:4 (2022), 14–27 | DOI | MR