A functional limit theorem for the profile of random recursive trees

Let Xn(k) be the number of vertices at level k in a random recursive tree with n + 1 vertices. We are interested in the asymptotic behavior of Xn(k) for intermediate levels k = kn satisfying kn → ∞ and kn = o(log n) as n → ∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn([knu])) u>0 , properly normalized and centered, as n → ∞. The limit is a centered Gaussian process with covariance (u, v) → (u + v) −1 . One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs, Hwang and Neininger in [5]. In contrast, our proofs which are probabilistic in nature exploit a connection of our model with certain Crump-Mode-Jagers branching processes.

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“…The present article is a follow-up of [9] in which a functional limit theorem was proved for the random process X [n u ] (1), . .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proof. Without any restrictions on the distribution of a positive random variable ξ the following formulas were obtained in Lemma 4.2 of [9]: for k ≥ 2 and t ≥ 0…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Weak convergence of the number of vertices at intermediate levels of random recursive trees

Iksanov

Kabluchko

2018

J. Appl. Probab.

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“…In this section we find, for fixed j ∈ N, the asymptotics of Var N j (t) as t → ∞ under the assumption η = ξ a.s., so that T k = S k for k ∈ N. In other words, below we treat iterated standard random walks. Theorem 2.6 is a strengthening of Lemma 4.2 in [7] in which the big O estimate for Var N j (t) was obtained, rather than precise asymptotics. We do not know the asymptotic behavior of Var N j (t) for (genuine) iterated perturbed random walks.…”

Section: Asymptotics Of the Variancementioning

confidence: 94%

Renewal theory for iterated perturbed random walks on a general branching process tree: early generations

Iksanov¹,

Rashytov²,

Самойленко³

2021

Preprint

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Let (ξ k , η k ) k∈N be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequenceConsider a general branching process generated by T and denote by N j (t) the number of the jth generation individuals with birth times ≤ t. We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for EN j of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for N j , find the first-order asymptotics for the variance of N j . Also, we prove a functional limit theorem for the vector-valued process (N 1 (ut), . . . , N j (ut)) u≥0 , properly normalized and centered, as t → ∞. The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

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“…To check (5) we assume for simplicity that σ 2 := Var ξ < ∞ (this condition is by no means necessary but enables us to avoid some additional calculations). Theorem 1.3 in [8] entails that…”

Section: Positions In the Jth Generation Of A Branching Random Walkmentioning

confidence: 99%

“…We refer to [6] and [10] for detailed surveys concerning earlier works on random processes with immigration at the epochs of a Poisson or renewal process. A non-exhaustive list of more recent contributions, not covered in the cited sources, includes [7], [8], [9], [12] and [13].…”

Section: Introductionmentioning

confidence: 99%

Weak convergence of random processes with immigration at random times

Dong,

Iksanov

2019

Preprint

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By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. The so defined random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in [Iksanov et al. (2017). Bernoulli, 23, 1233-1278 and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

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A functional limit theorem for the profile of random recursive trees

Cited by 7 publications

References 19 publications

Weak convergence of the number of vertices at intermediate levels of random recursive trees

Weak convergence of the number of vertices at intermediate levels of random recursive trees

Renewal theory for iterated perturbed random walks on a general branching process tree: early generations

Weak convergence of random processes with immigration at random times

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