Suppose $\{X_i\}_{i = 0}^\infty$ is a Galton-Watson branching process ($\{X_i\}_{i = 0}^\infty$ is a sequence of random variables satisfying the conditions $P(X_0)= 1$ and $P(X_i) = \Sigma_{i = 1}^{X_{i - 1}} \theta_ij \forall i \in \mathbb{N}$ for some sequence of i.i.d random variables $\{\theta_ij\}_{i, j \in \mathbb{N}}$). Define $\alpha_n := \inf\{t \in \mathbb{R}: (E\theta_i < t) \to \exists N \in \mathbb{N} (P(\exists N \in \mathbb{N} \forall k > N X_k < n) = 1)\}$.
Is there some sort of a closed formula for $\alpha_n$?
I know, that $\forall n \in \mathbb{N} \alpha_n \geq 1$, as $(E\theta_i < t) \to (P(\exists N \in \mathbb{N} \forall k > N X_k = 0) = 1)\}$. However, that does not solve my problem.