Scaling limit of multitype Galton–Watson trees with infinitely many types

Abstract

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton-Watson trees with a countable number of types, thus extending a result of G. Miermont in [18] on multitype Galton-Watson trees with finitely many types.MSC2010 subject classification 60J80, 60F17

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

“…Moreover, under an additional exponential moment assumption, he also established that conditionally on the number individuals of a given type, the limit is given by the normalized Brownian excursion. More recently, de Raphelis [8] has extended the unconditional result in [21] for multitype GW trees with infinitely many types, under similar assumptions. Informally speaking, these results claim that multitype GW trees behave asymptotically in a similar way as the monotype ones, at least in the finite variance case.…”

Gromov–Hausdorff–Prokhorov convergence of vertex cut-trees of $n$-leaf Galton–Watson trees

“…Let us be precise and extend the planar order of G (n) to G (n) by placing all extra children to the left. Following ideas of Miermont [37] and de Raphélis [17], we introduce the following notation. Let u(i) = v i ( G (n) ), i = 0, 1, .…”

Scaling limit of the recurrent biased random walk on a Galton–Watson tree

“…Let us introduce the result of [8] that will be used in our proof. Let T be a leafed Galton-Watson tree with edge lengths (as defined in Section 1.1 of [8]) ; that is T is a 2-type Galton-Watson tree with edge lengths with types denoted by s and f such that vertices of type s give no offspring (they are sterile).…”

“…trees (T i ) i distributed as T and we call H the height function associated to the forest, and H f the height function of the forest restricted to vertices of type f . The following result comes from Theorem 1 of [8] (beware that H f is different from the process H 1 introduced in [8] since in our case the lengths are not reset to 1). It states that the height function H is asymptotically given by a deterministic rescaling in time of H f .…”

“…In Section 4, we construct different reduced trees associated to the trace of an excursion. These reduced trees are simpler to deal with since they fall into the scope of [8], but still contain all the information needed. We prove Theorem 1.1 in Section 5.…”