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Consider a branching random walk given by the collection of i.i.d random variables $X(i_{0},\ldots,i_{t})$. Here $t \in \mathbb{N}$ and $i_{k} \in \{1,\ldots,n\}$ for any $k \in \mathbb{N}$. Each $X(i_{0},\ldots,i_{t})$ is Bernoulli, taking value 0 with probability $p$ and 1 with probability $1-p$. Consider the maximal displacement of the BRW at time $t$: $$M_{t} = \max_{(i_{0},\ldots,i_{t})}\sum_{k = 0}^{t} X(i_{0},\ldots,i_{k}).$$ I am interested in the asymptotic behaviour of $M_{t}/t$. Specifically:

(1) Is it correct that $\tfrac{M_{t}}{t} \to 1$ a.s. if $p < 1-\tfrac{1}{n}$? (My logic is to apply the textbook percolation result. In fact, it even seems to tell me more: a.s. there is a $t_{0} \in \mathbb{N}$ such that $M_{t} \geq t-t_{0}$ for all $t \geq t_{0}$.)

(2) Is it correct that the previous conclusion is still true if $p = 1-\tfrac{1}{n}$?

(3) What can I say if $1-\tfrac{1}{n} < p$?

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    $\begingroup$ I think there is a significant number of typos, please correct. For example, the sum in the RHS of the definition of $M_t$ does not have $t$ in it, so how do you take the indicated max? Also, this seems not to be a BRW. Finally, how do you couple the different r.v.'s when you change $n$ (in order to discuss a.s. behavior)? Depending on the answer, the claim could be true for and $p>0$, and $p>p_c$ with $p_c<1$, or just false. $\endgroup$ Commented Sep 29 at 21:56
  • $\begingroup$ @oferzeitouni: thanks for your comment! typo corrected. My use of the term "BRW" seems to agree with that in Bramson MD: Maximal displacement of branching Brownian motion. Communications on Pure and Applied Mathematics. 1978 Sep;31(5):531-81. As for the question of "coupling": I suppose we may think of all the increments defined on the same probability space, for instance the product space $\{0,1\}^{T}$ where $T$ is the complete $n$-ary tree. $\endgroup$ Commented Sep 30 at 11:06
  • $\begingroup$ Now that $n$ is not a parameter in the limit, the question is well defined. $\endgroup$ Commented Oct 1 at 1:01
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    $\begingroup$ Look at Theorem 1.3 of Zhan Shi's St Flour course (Springer LNM 2151). After obvious change of notation and sign change, you will see that if $n(1-p)\geq 1$ then (in his notation) the infimum of $\psi(t)/t$ is obtained at $t\to \infty$, which gives velocity=1, while otherwise the velocity is the infimum of $\log(np+n(1-p)e^t))/t$ over $t>0$. $\endgroup$ Commented Oct 1 at 1:52
  • $\begingroup$ @oferzeitouni: thanks! question answered. $\endgroup$ Commented Oct 5 at 14:00

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Look at Theorem 1.3 of Zhan Shi's St Flour course (Springer LNM 2151). After obvious change of notation and sign change, you will see that if $n(1−p)≥1$ then (in his notation) the infimum of $ψ(t)/t$ is obtained at $t\to \infty$, which gives velocity=1, while otherwise the velocity is the infimum of $\log(np+n(1−p)e^t))/t$ over $t>0$.

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