For a particular case of a branching random walk with lattice support, namelythe Yule branching random walk, we prove that the distribution of the centredmaximum oscillates around a distribution corresponding to a critical travellingwave in the following sense: there exist continuous functions $t \mapsto a_t$and $x \mapsto \overline{\phi}(x)$ such that: $$\lim_{t \rightarrow +\infty} \sup_{x \in \mathbb{R}} \vert\mathbb{P}(\overline{X}(t) \leq a_t +x )-\overline{\phi}(x- \{ a_t+x\})\vert=0,$$ where $\{x\}=x-\lfloor x \rfloor$ and $\overline{X}(t)$ is theheight of the Yule tree. We also shows that similar oscillations occur for$\mathbb{E}\left(f(\overline{X}(t)-a_t)\right)$, when $f$ is in a large classof functions. This process is classically related to the binary search tree,thus yielding analogous results for the height and for the saturation level ofthe binary search tree.
展开▼
机译：对于带有格支撑的分支随机游动的特殊情况，即Yule分支随机游走，我们证明在以下意义上，中心极大值的分布围绕着与临界行波相对应的分布振荡：存在连续函数$ t \ mapsto a_t $和$ x \ mapsto \ overline {\ phi}（x）$这样：$$ \ lim_ {t \ rightarrow + \ infty} \ sup_ {x \ in \ mathbb {R}} \ vert \ mathbb {P} （\ overline {X}（t）\ leq a_t + x）-\ overline {\ phi}（x- \ {a_t + x \}）\ vert = 0，$$其中$ \ {x \} = x- \ lfloor x \ rfloor $和$ \ overline {X}（t）$是Yule树的高度。我们还表明，当$ f $在一大类函数中时，$ \ mathbb {E} \ left（f（\ overline {X}（t）-a_t）\ right）$也会发生类似的振荡。这个过程通常与二叉搜索树相关，因此对于二叉搜索树的高度和饱和度会产生相似的结果。
展开▼
