Suppose $\{X_{i, j}\}_{i, j \in \mathbb{N}} \sim X$ are i.i.d. random variables from $\mathbb{Z}$. Let's define a sequence of random variables $\{B_n\}_{n \in \mathbb{N}}$ defined by the following recurrence:
$$B_1 = X_{1, 1}$$
$$B_{n+1} = \sum_{i = 1}^{|B_n|} X_{n+1, i}$$
Let's define $\phi(X) := P(\exists n \in \mathbb{N} B_n = 0)$.
Suppose $|E(X)| \leq 1$ and $X$ is non-constant. Is it always true, that $\phi(X) = 1$?
It is true for the particular cases $P(X \geq 0) = 1$ (case 1) and $P(X \leq 0) = 1$ (case 2) because both of them are equivalent to ordinary branching processes. However, I do not know, how to deal with this problem in general.