I am looking for any known results about the tail sigma-algebra of a branching random walk. To be specific, let $T$ be the nodes of an infinite binary tree rooted at $r \in T$. Let $\{X_t\})_{t \in T}$ be i.i.d. real random variables with distribution $\mu$. For $s \in T$, let $Y_s$ equal to the sum $\sum_{t \in P_s}X_t$, where $P_s \subset T$ is the set of nodes on the unique path from the $r$ to $s$.

Let $\mathcal{F}_n$ be the sigma-algebra generated by all the $Y_s$ that are at distance at least $n$ from $r$, and let the tail sigma-algebra be given by $\mathcal{F}_\infty = \cap_n\mathcal{F}_n$.

Is the tail sigma-algebra trivial? Does it depend on $\mu$? Anything known about specific cases, for example when $\mu$ is Gaussian?