Limit distributions of the number of vertices in strata of a simply generated forest

We consider simply generated forests consisting of N trees and n non-root vertices. As N,n -> °° so that N/n —>· «>, we obtain the limit distributions of the number of vertices in strata of these forests. This research was supported by the Russian Foundation for Basic Research, grant 97-01-00065. The asymptotic behaviour of the number of vertices in strata of a random forest was first discussed in [1, 2]. The forests considered in these papers consist of N rooted trees with labelled vertices, whose roots are labelled with 1,2,. ,W, and non-root vertices have labels 1,2,. ,n. The uniform probability distribution is given on the set of all these forests. In addition, in [1] it is assumed that the degrees of the vertices take values from some subset R of non-negative integers with zero (the degree of a vertex is the number of vertices emerging from it, provided that the arcs are directed away from the roots). We say that the height of a vertex is the number of arcs that form the path from the root to this vertex, and let the ith stratum of the forest be the set of vertices of height t. In [1], limit distributions were obtained for the number of vertices in the tth stratum for two cases of grow of W and n. In the former case, N = cn(l + o(l)), 0 < c < l, t is either fixed or t ->· °o, n(l cj -> °o; in the latter case N = O(\/n), t/^/n —>· 0. In [2], similar limit theorems were proved for n —> °o? N/^/n<C\ < oo, t/y/fi < €2 < 0°. Under these constraints, the height of a random forest (the maximal number of a stratum) is of order ^/n (see [3]); therefore, the results of [2] give us an exhaustive description of the behaviour of the number of vertices in strata of a forest in the domain of variation of parameters specified above. The idea of proofs in [1, 2] goes back to the known monograph [4] due to V. F. Kolchin, where the asymptotic behaviour of the number of vertices in strata of a random tree with n labelled vertices was studied as n —t oo. in [4], this problem was reduced to consideration of the critical Galton-Watson branching process starting from a single particle, with the Poisson-distributed offspring under the condition that the progeny is equal to n. It is natural that to study the above-mentioned random forests, in [1, 2] a similar branching process was used starting from TV particles. *UDC 519.2. Originally published in Diskretnaya Matematika (1999) 11, No. 1, 97-112 (in Russian). Received December 6, 1998. Translated by A. V. Kolchin. 138 Yu. L. Pavlov and I. A. Cheplyukova In [5], results similar to those of [2] were obtained for forests consisting of TV plane planted trees and n non-root vertices under the uniform probability distribution given on the set of these forests. In this case, to deduce limit theorems the conditional Galton-Watson branching process with geometric-distributed offspring was used. In [6, 7], forests consisting of N simply generated trees were studied. The definition of a simply generated family of rooted trees was given in [8]. It is known (see, e-g[9]) that each simply generated family can be considered as a set of realizations of some Galton-Watson branching process starting from a single particle. Similarly, a branching process G starting from TV particles yields a simply generated forest of N trees and in a natural way induces on it some probability distribution. Let J^ denote the set of all these forests with n non-root vertices. From [8, 9] it follows that the random forests considered in [1, 2, 3, 5] are particular cases of J^v,«· Let χ be some numerical characteristic of a random forest from J^,«» %(G) be the analogous characteristic of the corresponding branching process G. Let v^ denote the random variable equal to the progeny of the process G. In [7], the following assertion was proved. Lemma 1. The equality is true. This lemma demonstrates that the study of numerical characteristics of a simply generated forest reduces to analysis of the corresponding characteristics of the branching process G under the condition v^=N + n. Lemma 1 extends Lemma 1 . 4. 1 of [3], therefore all results of [3] concerning random forests are valid for ^v>n. Let μ(ί , c^)v,/i) stand for the number of vertices in the fth stratum of the random forest from ^γ,/ι» d let μ#(0 be ^e number of particles of the rth generation of the branching process G. Lemma 1 implies that the distributions of ju(f,G#,n) and are related as follows. Lemma 2. For any non-negative integers χ and t, the equality Ρ{μ(ί,&Ν^)=χ} = Ρ{μΝ(ή=χ\ νΝ=Ν + η} is true. In the case where the variance of the offspring is finite, in [6] a complete description of the limit behaviour of μ(ί , J^,n) was given as η — >· °°, Ν /^/n <C\ < <». In this paper, we consider the asymptotic behaviour of this random variable in other domains of variation of the parameters TV, n. The strata numbers t in the results below, roughly speaking, satisfy the relation t — ο (τ), where τ is the forest height (the limit distributions of the forest height for all kinds of grow of Ν and η were obtained in [3]). Limit distributions of the number of vertices in strata of a forest 139 We introduce the auxiliary random variable ξ, Ρ{ξ = *} = /*, £ = 0,1,2, . , with span d and the generating function *=o the set of values of ξ contains zero with a positive probability and does not coincide with the set {0, 1}. In addition, we assume that Εξ = 1, Οξ = Β < <~, F'"(l) < οο. It is well known (see, e.g. [9]) that, in the case where the first three moments of the offspring distribution in the branching process G that generates &Ν,Π are finite, this distribution can be defined as follows: * = 0,1,2, . , λ>0. (1) In what follows we assume that the parameter λ = λ(Ν,η) is determined by the relation λ^(λ) = η Ρ(λ) N + n ^ } Let m and Βχ denote the mathematical expectation and variance of distribution (1) respectively. From (2) it follows that m — n/(N + n). The assertions below are true. Theorem 1. Let N,n -> <~ so that n/N < C < oo, Nmf ->· oo. Let t > 3 as n/N — ϊ 0 and t — > <» otherwise. Then