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

Conditional transfer to weighted planar-map blocks

Conditional transfer to weighted planar-map blocks · lines 1344–1364

Oai211-A@1#eq:blockoffspring

This is the endpoint argument reported in OpenAI211 (Lemma 4.2), included locally. It requires no analytic continuation across $v$ .

Define

p_{2 m} = \frac{b _{m} v ^{m}}{Φ ( v )}, p_{2 m + 1} = 0.

Then

E ξ = 1, σ^{2} = 1 + \frac{4 v ^{2} Φ ^{''} ( v )}{Φ ( v )} \in (0, \infty) .

At a node with $2 m > 0$ children, choose block $B$ with probability $w (B) / b_{m}$ . Additional operational marks may be chosen independently at nodes, conditional on their selected blocks and offspring numbers.

Displayed formulas in this passage

Oai211-A@1#eq:blockoffspring · line 1351
$p_{2 m} = \frac{b _{m} v ^{m}}{Φ ( v )}, p_{2 m + 1} = 0.$
unlabeled · line 1356
$E ξ = 1, σ^{2} = 1 + \frac{4 v ^{2} Φ ^{''} ( v )}{Φ ( v )} \in (0, \infty) .$

Nearby in this section

section: Conditional transfer to weighted planar-map blocks
definition: Definition: Weighted corner-substitution contract
section: Conditional transfer to weighted planar-map blocks
assumption: Assumption: Endpoint coefficient inputs
lemma: Lemma: Quadratic inverse criterion
theorem: Theorem: Exact conditional weighted representation
section: Conditional transfer to weighted planar-map blocks