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?



Tail sigma-algebra of a branching random walk
Actually, your idea of looking at the generation sums can be made more explicit. Let $s\_n$ be the sum of $X\_t$ over all vertices $t$ at a level $n\ge 0$, and let $S\_n$ be the corresponding sum of $Y\_t$ (in particular, $s\_0=S\_0=X\_r$). Then
$$
S\_n = 2^n s\_0 + 2^{n-1} s\_1 + \dots + 2 s\_{n-1} + s\_n \;,
$$
or 
$$
2^{-n} S\_n = s\_0 + \frac{s\_1}2 + \dots + \frac{s\_{n-1}}{2^{n-1}} + \frac{s\_n}{2^n} \;.
$$
Therefore, at least under the assumption that $\mu$ has a finite second moment,
$2^{-n} S\_n - n \overline\mu$ (where $\overline\mu$ is the expectation of $\mu$) converges a.s. to a non-trivial limit .


Unfortunately, this construction leaves quite a number of natural questions completely open.


