Tree-tail asymptotics and the tail-to-exact ratio #
The lattice mass asymptotic is summed with the exact residue-class Karamata constant. The resulting tree-tail asymptotic holds along all integers.
Reference: J.-F. Le Gall, Random trees and applications, Probab. Surveys 2 (2005), Section 1.
theorem
BGW.StableTrees.treeTail_asymp
{μ : PMF ℕ}
{θ : ℝ}
{B : ℕ → ℝ}
(h : StableDomain μ θ B)
:
Filter.Tendsto (fun (n : ℕ) => B n * forestTail μ 1 n) Filter.atTop (nhds (θ * Scalar.stableDensity θ 0))
Tree tail asymptotics along ALL n: B_n G_1(n) → θ p₁(0). No admissibility needed.
theorem
BGW.StableTrees.tailExactRatio_tendsto
{μ : PMF ℕ}
{θ : ℝ}
{B : ℕ → ℝ}
(h : StableDomain μ θ B)
{q : ℕ → ℕ}
(hqtop : Filter.Tendsto q Filter.atTop Filter.atTop)
(hq : ∀ (k : ℕ), Admissible μ (q k))
:
Filter.Tendsto (fun (k : ℕ) => forestTail μ 1 (q k) / (↑(q k) * forestMass μ 1 (q k))) Filter.atTop
(nhds (θ / ↑(Scalar.latticeSpan μ)))
Tail/exact ratio along admissible sizes: G_1(q_k) / (q_k f_1(q_k)) → θ / d.