Geodesic
Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants
Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 19-31

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We consider the branching process $Z_{n} = X_{n, 1} + \dotsb + X_{n, Z_{n-1}}$ in random environments $\boldsymbol\eta$, where $\boldsymbol\eta$ is a sequence of independent identically distributed variables and for fixed $\boldsymbol\eta$ the random variables $X_{i,j}$ are independent and have the geometric distribution. We suppose that the associated random walk $S_n = \xi_1 + \dotsb + \xi_n$ has positive mean $\mu$ and satisfies the right-hand Cramer's condition ${\mathbf E}\exp(h\xi_i) \infty$ for $0$ and some $h^{+}$. Under these assumptions, we find the asymptotic representation for local probabilities ${\mathbf P}\left( Z_n = \lfloor\exp\left(\theta n\right)\rfloor \right)$ for $\theta \in [\theta_1,\theta_2] \subset (\mu;\mu^+)$ and some $\mu^+$.
Keywords: branching processes, random environments, random walks, Cramer's condition, large deviations, local theorems. 
K. Yu. Denisov. Asymptotic local probabilities of large deviations for branching process in random environment with geometric distribution of descendants. Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 19-31. http://geodesic.mathdoc.fr/item/DM_2021_33_4_a2/
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[1] Kozlov M. V., “On large deviations of branching processes in a random environment: geometric distribution of descendants”, Discrete Math. Appl., 16:2 (2006), 155–174 | DOI | DOI | MR | Zbl

[2] Kozlov M. V., “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 | DOI | MR | Zbl

[3] Bansaye V., Berestycki J., “Large deviations for branching processes in random environment”, Markov Process. Related Fields, 15:3 (2009), 493–524 | Zbl

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

[5] Shklyaev A.V., “Bolshie ukloneniya vetvyaschegosya protsessa v sluchainoi srede. II”, Diskretnaya matematika, 32:1 (2020), 135–156 | DOI | MR

[6] Borovkov A.A., Asimptoticheskii analiz sluchainykh bluzhdanii. Bystroubyvayuschie raspredeleniya priraschenii, Fizmatlit, 2013, 447 pp.

[7] Petrov V.V., “On the probabilities of large deviations for sums of independent random variables”, Theory Probab. Appl., 10:2 (1965), 287–298 | DOI | MR | Zbl

[8] Agresti A., “On the extinction times of varying and random environment branching processes”, J. Appl. Prob., 12:1 (1975), 39–46 | DOI | Zbl

[9] Denisov K. Yu., “Asimptotika lokalnykh veroyatnostei nizhnikh uklonenii vetvyaschegosya protsessa v sluchainoi srede pri geometricheskikh raspredeleniyakh chisel potomkov”, Diskretnaya matematika, 32:3 (2020), 24–37 | DOI | MR