Conference Agenda

We consider a last progeny modified branching random walk, in which the position of each particle at the last generation $n$ is modified by an i.i.d. copy of a random variable $Y$ . Depending on the asymptotic properties of the tail of $Y$ , we describe the asymptotic behaviour of the extremal process of this model as $n\to\infty$.

We consider random walks on supercritical Galton-Watson trees with random conductances. That is, given a Galton-Watson tree, we assign to each edge a positive random weight (conductance) and the random walk traverses an edge with a probability proportional to its conductance. On these trees, the random walk is transient and the distance of the walker to the root satisfies a law of large numbers with limit the speed of the walk. We show that the distance of the walker to the root satisfies a functional central limit theorem under the annealed law. When a positive fraction of edges is assigned a small conductance $\varepsilon$, we study the behavior of the limiting variance as $\varepsilon\to 0$. Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by the $\varepsilon$-edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in $\varepsilon$.

It is well known that a random walk $S_n = \sum_{k=1}^n X_k$, with $(X_k)$ i.i.d. having finite expecation diverges almost surely to $\infty$ if and only if $E (X_1) > 0$, while for $E (X_1) = 0$ it oscillates. Less well known is the study of random walks when $E |X_1| = \infty$. In 1973, Erickson [1] obtained an integral criterion characterising when the corresponding random walk diverges to $\infty$, $-\infty$, or when it oscillates. In this talk we are interested in the divergence behaviour of $W_n=\sum_{k=1}^n c^k X_k$, where $(X_k)$ is i.i.d. and $0<c<1$. It is well known that this sum converges almost surely if and only if $E \log^+ |X_1| < \infty$, but we are interested in the divergence behaviour when $E \log^+ |X_1| = \infty$. We give sufficient analytic conditions for $W_n$ to exhibit an almost sure oscillating behaviour (i.e. $-\infty = \liminf\limits_{n\to\infty}W_n<\limsup\limits_{n\to\infty}W_n=\infty$), as well as a sufficient criterion for the almost sure limit to exist in the sense that $\lim\limits_{n\to\infty}W_n=\infty$. The talk is based on joint work in progress with A. Lindner and R. Maller.

[1] K. Bruce Erickson. “The strong law of large numbers when the mean is undefined”. In: Transactions of the American Mathematical Society 185 (1973), pp. 371–381.

The Ewens-Pitman model is a distribution for random partitions of $\{1,\ldots,n\}$, with $n\in\mathbb{N}$, indexed by a pair of parameters $\alpha \in [0,1)$ and $\theta>-\alpha$, such that $\alpha=0$ corresponds to the Ewens model in population genetics. The large $n$ asymptotic behaviour of the number $K_{n}$ of blocks in the Ewens-Pitman random partition has been extensively investigated in terms of almost-sure and Gaussian fluctuations, which show that $K_{n}$ scales as $\log n$ or $n^{\alpha}$ depending on whether $\alpha=0$ or $\alpha\in(0,1)$, providing non-random and random almost-sure limits, respectively.

We study the large $n$ asymptotic behaviour of $K_{n}$ when the parameter $\theta$ is allowed to depend linearly on $n\in\mathbb{N}$. Precisely, for $\alpha\in[0,1)$ and $\theta=\lambda n$, with $\lambda>0$, we establish a law of large numbers (LLN) and a central limit theorem (CLT) for $K_{n}$, which show that $K_{n}$ scales as $n$ for $\alpha\in[0,1)$, providing a non-random almost sure (a.s.) limit. In particular, for $\alpha\in(0,1)$, the CLT relies on the compound Poisson construction of $K_{n}$, which leads to introduce novel LLNs, CLTs and corresponding Berry-Esseen theorems for the negative-Binomial compound Poisson random partition and for the Mittag-Leffler distribution function, which are of independent interest.

In conclusion, we show an application of our results to the problem of uncertainty quantification in the Bayesian nonparametric (BNP) approach to the estimation of the number of unseen species. Given $n\geq1$ observed individuals, modeled as a random sample $(X_{1},\ldots,X_{n})$ from the two-parameter Poisson-Dirichlet distribution, the unseen-species problem calls for estimating the number $K_{m}^{(n)}$ of hitherto unseen distinct species that would be observed if $m\geq1$ additional samples were collected from the same distribution. The posterior expectation of $K_{m}^{(n)}$ (i.e. its conditional distribution given the Ewens-Pitman random partition induced by $(X_{1},\ldots,X_{n})$) is the natural BNP estimator of $K_{m}^{(n)}$.

For $\alpha\in[0,1)$ and in the regime $m = \lambda \theta$, we develop for $K_{m}^{(n)}$ posterior counterparts of the LLN and CLT for $K_n$. This allows to introduce large $m$ Gaussian credible intervals for the Bayesian estimator of $K_{m}^{(n)}$, whose construction is purely analytical and which outperform the existing methods.