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Take a continuous-time Galton-Watson tree and pick k distinct particles uniformly from those alive at a time T . What does their genealogical tree look like? The case k = 2 has been studied by several authors, and the near-critical asymptotics for general k appear in Harris, Johnston and Roberts (2018) [9]. Here we give the full picture. IntroductionLet L be a random variable taking values in {0, 1, 2, . . .}. Consider a continuous-time Galton-Watson tree starting with one initial particle, branching at rate 1, and with offspring distributed like L. Let N t be the number of particles alive at time t, and write f (s) := E[s L ] and F t (s) := E[s Nt ] for the generating functions associated with the process.Let T > 0, and on the event {N T ≥ k} pick k distinct particles U 1 , . . . , U k uniformly from those alive at time T . For each earlier time t ∈ [0, T ], define the equivalence relation ∼ t on {1, . . . , k} by i ∼ t j ⇐⇒ U i and U j share a common ancestor alive at time t.We let π k,L,T t denote the random partition of {1, . . . , k} corresponding to this equivalence relation. The process (π k,L,T t ) t∈[0,T ] , defined on the event {N T ≥ k}, is a right-continuous partition-valued stochastic process characterising the entire genealogical tree of U 1 , . . . , U k .Our goal is to describe the law of (π k,L,T t ) t∈[0,T ] conditioned on the event {N T ≥ k}, with a view towards the asymptotic regime T → ∞. We find that as T → ∞, there are marked differences in the qualitative behaviour of (π k,L,T t ) t∈[0,T ] depending on the mean number of offspring m := f ′ (1).Before we state our results in full generality in Section 3, we give an impression of the structure we expect to encounter by exploring the special case k = 2, which features as the focus of a chapter in the recent book [3], and on which the majority of the related literature concentrates. The case k = 2The case k = 2 amounts to choosing two particles uniformly from those alive at a time T from a tree with offspring distributed like L, and studying the time τ L,T in [0, T ] at which they last shared a common ancestor. In terms of the partition process (π 2,L,T t ) t∈[0,T ] , τ L,T is the time at which the single block {{1, 2}} splits into the pair of singletons {{1}, {2}}.The following characterisation of the law of τ L,T (which we will generalise later) was first given by Lambert [14].

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In [A dozen de Finetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist. 23(2) (1987), 397-423], Diaconis and Freedman studied the low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Rüschendorf and Naor and Romik unified these results by establishing a connection between ℓ N p balls and a p-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the formwhere φ : R → [0, ∞] is a function satisfying some fairly mild conditions; in particular, we cover the case of Orlicz functions. Our method is different from both Rachev-Rüschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cramér's theorem and the Gibbs conditioning principle, providing a natural framework beyond the pgeneralized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of ℓ N p balls. We find that there is a critical parameter t crit at which there is a phase transition in the behaviour of the low-dimensional projections: for t > t crit the coordinates of random vectors sampled from B N φ,t behave like uniform random variables, but for t ≤ t crit however the Gibbs conditioning principle comes into play, and here there is a parameter β t > 0 (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to e −β t φ(s) .

In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an ℓ n p -ball which has been obtained in [Alonso-Gutiérrez, Prochno, Thäle: Gaussian fluctuations for highdimensional random projections of ℓ n p -balls, Bernoulli 25(4A), 2019, 3139-3174]. More precisely, for any n ∈ N let En be a random subspace of dimension kn ∈ {1, . . . , n}, PE n the orthogonal projection onto En, and Xn be a random point in the unit ball of ℓ n p . We prove a Berry-Esseen theorem for PE n Xn 2 under the condition that kn → ∞. This answers in the affirmative a conjecture of Alonso-Gutiérrez, Prochno, and Thäle who obtained a rate of convergence under the additional condition that kn/n 2/3 → ∞ as n → ∞. In addition, we study the Gaussian fluctuations and Berry-Esseen bounds in a 3-fold randomized setting where the dimension of the Grassmannian is also chosen randomly. Comparing deterministic and randomized subspace dimensions leads to a quite interesting observation regarding the central limit behavior. In this work we also discuss the rate of convergence in the central limit theorem of [Kabluchko, Prochno, Thäle: High-dimensional limit theorems for random vectors in ℓ n p -balls, Commun. Contemp. Math. ( 2019)] for general ℓq-norms of non-projected vectors chosen at random in an ℓ n p -ball.