Tail sigma-algebra of a branching random walk

Question

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?

Answer 1

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.

Answer 2

It turns out that $\mathcal{F}_\infty$ is never trivial (at least when the distribution of $X_s$ has a second moment). This can be seen by calculating the covariance of the average $Y_s$ at generation $n$ with $Y_r$, and comparing it to the variance of this average.