Small deviations for critical Galton-Watson processes with infinite variance

TL;DR

The paper investigates small deviation probabilities for critical Galton–Watson processes with infinite offspring-variance, characterized by $f(s)=s+(1-s)^{1+\alpha}L(1-s)$ with $\alpha\in(0,1]$ and slowly varying $L$. It develops a framework based on Bell polynomials, slowly varying function analysis, and Tauberian arguments to obtain the asymptotic for $\mathbf{P}(\mathcal{H}(n))$ where $\mathcal{H}(n)=\{0<(1-f_{\varphi(n)}(0))Z(n)\leq1\}$ and $\varphi(n)/n\to0$, showing $\mathbf{P}(\mathcal{H}(n))\sim \frac{1-f_{n}(0)}{\alpha n}\varphi(n)$. The authors then apply these results to the reduced Galton–Watson process, proving a limit theorem for $\mathbf{P}(Z(n-x\varphi(n),n)=j|\mathcal{H}(n))$ in terms of $M^{*j}$, the $j$-fold convolution of the Yaglom limit, and deriving a corresponding MRCA-distance description. The work extends finite-variance results to infinite-variance settings, providing a precise description of the conditional genealogical structure under small deviations. Overall, the paper contributes new asymptotics and structural results for critical branching processes with heavy tails.

Abstract

We study the asymptotic behavior of small deviation probabilities for the critical Galton-Watson processes with infinite variance of the offspring sizes of particles and apply the obtained result to investigate the structure of a reduced critical Galton-Watson process.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Remark 3
• Corollary 4
• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2