Leaf-height declumping: the root formula and the conditional mark bound #
Section 7.2 of the paper (source lines 3132–3181, 3236–3241; inventory U11 and Lemma
lem:lh_conditional_mark). For the leaf-height marks G_k(T) = 1{λ_∅(T) ≥ k}, with
κ = min{j ≥ 1 : μ_j > 0}, the declumped mark is
Ghat_k(T) = 1{k_∅(T) ≥ κ + 1} 1{∃ u ∈ Γ_∅(T), λ_∅(T_u) ≥ k} (lhDeclumpedMark).
Write π_k = P(λ_∅(𝒯) ≥ k) (piSeq, identified by bienayme_real_le_rootLeafHeight) and
π̂_k = P(Ghat_k(𝒯) = 1).
- U11:
lh_declumped_prob_eq:π̂_k = ∑_{j ≥ κ+1} μ_j (1 - (1 - π_k)^j);lh_declumped_sub_isLittleO,lh_declumped_tendsto_div:π̂_k = c π_k + o(π_k),c = 1 - κ μ_κ;lh_declumped_isEquivalent:π̂_k ∼ c π_kwhenc > 0;bB_mul_le_lh_declumped:π̂_k ≥ b π_kwithb = P(k_∅ ≥ κ + 1)(bB_lhDegrees_eq), andb > 0whenc > 0(bB_lhDegrees_pos). - Lemma
lem:lh_conditional_mark(lem_lh_conditional_mark): oneC > 0withP(Ghat_k(𝒯) = 1, λ_∅(𝒯) ≥ m) ≤ C π_k π_mfor all0 ≤ m ≤ k. The proof computes the left side exactly by the branching property,∑_{d ≥ κ+1} μ_d (π_{m-1}^d - (π_{m-1} - π_k)^d), which gives the paper's union boundπ_k ∑_{d ≥ κ+1} d μ_d π_{m-1}^{d-1}; then criticality andπ_m ≥ μ_κ π_{m-1}^κ. The assumptionc > 0of the surrounding subsection is not needed.
The declumped leaf-height mark Ghat_k(T) = 1{k_∅(T) ≥ κ + 1} 1{∃ u ∈ Γ_∅(T), λ_∅(T_u) ≥ k}
(source lines 3132–3137, with G_k(T) = 1{λ_∅(T) ≥ k} and B = {j ≥ κ + 1}).
Equations
- One or more equations did not get rendered due to their size.
Instances For
c_B = c = 1 - κ μ_κ for B = {j ≥ κ + 1} (source line 3031).
U11, the leaf-height declumped root formula (source lines 3134–3141):
π̂_k = ∑_{j ≥ κ + 1} μ_j (1 - (1 - π_k)^j).
U11: π̂_k - c π_k = o(π_k) as k → ∞ (dominated convergence, source line 3142).
U11: π̂_k / π_k → c (source line 3142).
U11: if c = 1 - κ μ_κ > 0, then π̂_k ∼ c π_k (source line 3142).
b = P(k_∅(𝒯) ≥ κ + 1) is the constant b_B of B = {j ≥ κ + 1} (source line 3236).
If c > 0 then b = P(k_∅(𝒯) ≥ κ + 1) > 0 (source line 3236).
π̂_k ≥ b π_k with b = P(k_∅(𝒯) ≥ κ + 1) (source lines 3236–3241).
π̂_k ≤ c π_k ≤ π_k.
Lemma lem:lh_conditional_mark #
The event {Ghat_k(𝒯) = 1, λ_∅(𝒯) ≥ m + 1} in product form: root degree ≥ κ + 1, all
children with leaf-height ≥ m, some child with leaf-height ≥ k.
Lemma lem:lh_conditional_mark (source lines 3148–3181): for a critical μ with
μ_1 ≠ 1, there is one constant C > 0 such that, for all 0 ≤ m ≤ k,
P(Ghat_k(𝒯) = 1, λ_∅(𝒯) ≥ m) ≤ C π_k π_m. One can take C = μ_κ⁻¹. (The paper states it in a
subsection assuming c > 0; that assumption is not needed.)
Lemma lem:lh_conditional_mark with π_k = P(λ_∅(𝒯) ≥ k) written as a probability.