Large Deviations and Berry–Esseen Bounds for A Critical Galton–Watson Process

Shengli Liang () and Qi-Man Shao ()
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Shengli Liang: Southern University of Science and Technology
Qi-Man Shao: Southern University of Science and Technology

Journal of Theoretical Probability, 2025, vol. 38, issue 4, 1-30

Abstract: Abstract Let $$\{Z_n\}$$ { Z n } be a critical Galton–Watson process and $$S_{Z_n}:=\sum _{k=1}^{Z_n}X_k$$ S Z n : = ∑ k = 1 Z n X k be sums of i.i.d. random variables $$\{X_k\}$$ { X k } . In this paper, we study the asymptotic behavior of $$S_{Z_n}/Z_n$$ S Z n / Z n conditioned on $$\{Z_n>0\}$$ { Z n > 0 } . A “phase transition” in rates is identified for the large deviation, depending on whether $$\beta $$ β , the negative index of the regularly varying right tail of $$X_1$$ X 1 , is less than, equal to or greater than 2. The self-normalized large deviation is also established without any moment assumption on $$X_1$$ X 1 . Moreover, Berry–Esseen bounds for the Lotka–Nagaev estimator, namely $$Z_{n+1}/Z_n$$ Z n + 1 / Z n , are derived by Stein’s method. As a by-product, new rates of convergence for Yaglom’s theorem are obtained.

Keywords: Berry–Esseen bound; Critical Galton–Watson process; Exponential approximation; Large deviation; Lotka–Nagaev estimator; Stein’s method; 60J80; 60F10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01444-7

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