A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees

By Igor Kortchemski. 2011

49 citations recorded by the source corpus.

We give a simple new proof of a theorem of Duquesne, stating that the properly rescaled contour function of a critical aperiodic Galton–Watson tree, whose offspring distribution is in the domain of attraction of a stable law of index θ ∈ (1, 2], conditioned on having total progeny n, converges in the functional sense to the normalized excursion of the continuous-time height function of a strictly stable spectrally positive Levy process of index θ. To this end, we generalize an idea of Le Gall which consists in using an absolute continuity relation between the conditional probability of having total progeny exactly n and the conditional probability of having total progeny at least n. This new method is robust and can be adapted to establish invariance theorems for Galton–Watson trees having n vertices whose degrees are prescribed to belong to a fixed subset of the positive integers.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/aa7f3a1fad4c4c3f11eda90a0b26d073f8af9974

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

1. Igor Kortchemski
2. Konstantinos Panagiotou
3. Kerstin Weller
4. Benedikt Stufler

In Mathematical Physics

1. Igor Kortchemski
2. Konstantinos Panagiotou
3. Kerstin Weller
4. Benedikt Stufler
5. Douglas Rizzolo

In Statistics and Probability

1. Igor Kortchemski
2. Konstantinos Panagiotou
3. Kerstin Weller
4. Benedikt Stufler

Citations in the corpus

• Scaling Limits of Random Graphs from Subcritical Classes (2014, background): “e e = (e t) 0t1 denotes the Brownian excursion of duration one. This result is due to Aldous [5, Thm. 23], who stated it for aperiodic ospring distributions. See also Duquesne [20] and Kortchemski [31] for further extensions. Theorem 2.2 can be formulated as a convergence of random trees with respec”
• Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees (2015, background): “Duquesne [18] (see also [30]) showed that the convergence ( 1 Bn · W nt (tn), n Bn · H nt (tn) )”