Variance of maximum over a dyadic tree

Question

Let $T_n$ be the vertices of the dyadic tree with root $v_0$ and depth $n$, and let $X_v$, $v \in T_n$, be independent identically distributed random variables with mean zero and variance one. Set $Z_n = \max _\pi \sum_{v \in \pi} X_v$ where the maximum is over all branches $\pi$ starting from $v_0$ with length (number of vertices) $n+1$. Is there anything known on the order of growth in $n$ of the variance of $Z_n$?

Answer 1

It depends on more than the variance of the increments. If they have exponential tails of high enough order, the variance is of order $1$ (look up "branching random walks", or the lecture notes of Shi https://link.springer.com/book/10.1007/978-3-319-25372-5

If they are heavy tailed, the situation is more complicated and might depend on the tail regularity of the variables.