Manuscript Oai211-A@1 · Oai211-A.tex · section · 3f3e4f63

The marked tree and its spine law

The marked tree and its spine law · lines 224–259

Oai211-A@1#eq:offspring

Let $p = (p_{k})_{k \geq 0}$ be an offspring law satisfying

p_{k} = 0 (k odd), p_{0} p_{2} > 0, k \sum k p_{k} = 1, 0 < σ^{2} := k \sum (k - 1)^{2} p_{k} < \infty.

Let $T$ be the corresponding plane Galton–Watson tree, and $T_{N}$ its law conditioned on $∣ T ∣ = N$ . Write

π_{N} = P (∣ T ∣ = N) .

The attainable sizes are exactly the positive odd integers: every tree has even total offspring count, and full binary trees realize every odd size. Throughout, $N \to \infty$ means convergence through these sizes.

At each node $u$ , independently conditional on the tree, assign a mark $A_{u}$ with law $L_{k_{u}}$ , where $k_{u}$ is the number of children of $u$ . A mark may contain a block, a finite channel vector, or an auxiliary random variable. Different children at the same node may use correlated functions of that common mark.

For a strict ancestor $u$ of $v$ , let $i_{u} (v) \in {1, \dots, k_{u}}$ be the child leading toward $v$ . Define

path (v) = ((k_{u}, i_{u} (v), A_{u}) : u a strict ancestor of v) .

Let $H (v)$ be depth.

Displayed formulas in this passage

Oai211-A@1#eq:offspring · line 227
$p_{k} = 0 (k odd), p_{0} p_{2} > 0, k \sum k p_{k} = 1, 0 < σ^{2} := k \sum (k - 1)^{2} p_{k} < \infty.$
unlabeled · line 237
$π_{N} = P (∣ T ∣ = N) .$
unlabeled · line 254
$path (v) = ((k_{u}, i_{u} (v), A_{u}) : u a strict ancestor of v) .$

The marked tree and its spine law

Displayed formulas in this passage

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