# Abstracts for the Melbourne-Singapore Probability & Statistics Forum

Kostantin Borovkov -- The University of Melbourne

The exact asymptotics of the large deviation probabilities in the multivariate boundary crossing problem

Abstract: For a multivariate random walk with i.i.d. jumps satisfying the Cramer moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results from a paper by F. Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a recent series of papers by A. Borovkov and A. Mogulskii with new auxiliary constructions, which enable us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a corner" at the most probable hitting point". Joint work with Yuqing Pan.

Peter Braunsteins -- The University of Melbourne

Extinction in block lower Hessenberg branching processes with countably many types

Abstract: We consider the extinction events of a class of branching processes with countably many types which we refer to as block Lower Hessenberg branching processes (BLHBPs). These are multitype Galton-Watson processes with typeset $\mathcal{X} = \{0, 1, 2, \dots \}$ whose mean progeny matrices are of block Lower Hessenberg form. Our approach involves embedding a finite-type explosive Galton-Watson process in a varying environment in the original BLHBP and then establishing asymptotic relationships between the two processes.

Using this method we derive local and global extinction criteria, and study the set $S$ of fixed points of the progeny generating function. We conclude with a discussion of some open questions concerning this set.

Louis H. Y. Chen -- National University of Singapore

Andrew Barbour and Stein’s Method

Abstract: Andrew Barbour has made fundamental contributions to Stein’s method. In this talk, we will discuss these contributions and other work of his on Stein’s method.

Andrea Collevecchio -- Monash University

Random walks in a Markovian Environment

Abstract: We consider random walks which jumps to nearest vertices of a random tree. Its transitions are determined by a random environment that evolves as an additive Markov process on rays. We establish a sharp criterion for recurrence/transience. Joint work with Andrew Barbour.

Xiao Fang -- Chinese University of Hong Kong

Moderate Deviations for the Erlang-C System

Abstract: Consider the Erlang-C system with number of servers $n$ and offer load $R$ (the ratio of the customer arrival rate to the service rate). Suppose the system is in the Quality- and Efficiency-Driven regime, that is, $n=R+\beta \sqrt{R}$ for a fixed constant $\beta$. Let $W$ be a scaled customer count in the steady state of the system.
Let $Y_{S}$ have the stationary distribution of a state-dependent diffusion.

We prove that $\left|\frac{P(W\geq z)}{P(Y_S\geq z)}-1\right|\leq \frac{C}{\sqrt{R}}(1+\frac{z\wedge R}{\sqrt{R}}),\quad z\geq 0$ and $\left|\frac{P(W\leq -z)}{P(Y_S\leq -z)}-1\right|\leq \frac{C}{\sqrt{R}}(1+z+\frac{z^4}{\sqrt{R}}),\quad 0\leq z=O(R^{1/4})$ where $C$ is a constant.

These upper bounds are of smaller order than that using the stationary distribution of a constant diffusion for approximation. To prove our results, we develop Stein's method for moderate deviations for higher order approximations.

This is joint work with Anton Braverman and Jim Dai.

Han Liang Gan -- Northwestern University

Approximating the difference of two Poisson-like counts

Abstract: Given a true random network graph, suppose we actually observe an estimated graph, which will have some measurement errors that take the form of extra or missing edges. As a simple example, suppose we wished to make inference on the number of edges in the true graph based upon our observed graph. Similarly to how inference upon a population mean requires knowledge of the distribution of the observed mean minus the true mean, in our scenario we will need to know the distribution, or at least the asymptotic distribution, of the difference in edge counts. Given we have count data, it is not unreasonable to suppose that this difference in counts could be well modelled by the difference of two independent Poisson random variables, also known as the Skellam distribution. To examine under what scenarios lead to a good Skellam approximation, we formulate Stein's method for the Skellam distribution. Our approach will involve using Barbour's generator approach and adapting bivariate Poisson approximation by considering test functions that depend only upon the difference of the two elements. This is joint work with Eric Kolaczyk.

Alexander Gnedin -- Queen Mary, University of London

Infinite random orders and permutations

Abstract: The obvious correspondence between permutations and (linear) orders on a finite set breaks when the ground set is $\mathbb N$, with permutations (bijections) $\mathbb{N} \to \mathbb{N}$ constituting a proper subclass of all orders.

In general, a random order on $\mathbb{N}$ can be represented as a permutation growth process, where integers are successively inserted in the one-row notation according to some random rule. We review some constructions of the kind, in particular that of Mallows permutations and size-biased permutations. A problem arises about the type of the order that appears in the limit. Addressing a question in Arratia, Barbour and Tavaré (2005), we identify possible types of the infinite size-biased orders.

Sophie Hautphenne -- The University of Melbourne

Parameter estimation for continuous-time branching processes observed at discrete times

Abstract: We develop an MLE method based on saddlepoint approximation to estimate the parameters of a linear birth-and-death process whose population size is observed at discrete times. We illustrate our results with numerical examples, and highlight the efficiency of our approach when compared to other existing moment based techniques.

Liam Hodgkinson -- The University of Queensland

Normal approximations for binary weakly interacting particle systems

Abstract: Growing interest in binary interacting particle systems, whose entities are treated as nodes with a binary state, has led to the development of general and highly detailed models of use in a wide variety of fields, including epidemiology, ecology, physics, and social and computer science. A common, and often very reasonable, assumption in these models is that particles transition individually. In this case, models with sufficiently weak interactions belong to a 'Gaussian universality class', and may be well approximated in some sense by a Gaussian process. In this talk, I will characterize this class of models by presenting, under no additional structural assumptions, an estimate on the rate of convergence in law of linear functionals of these processes to that of a Gaussian process, obtained using Stein's method.

Ross McVinish -- University of Queensland

Fast simulation of metapopulation models

Abstract: Tau leaping is a popular method for performing fast approximate simulation of certain continuous time Markov chain models typically found in chemistry and biochemistry. This method is known to perform well when the transition rates satisfy some form of scaling behaviour. In a similar spirit to tau leaping, we propose a simulation method for continuous time Markov chains that occur in modelling spatially structured biological populations, with metapopulations being the prototypical case. Sufficient conditions for the method to perform well will be presented and computational considerations discussed. This is joint work with Liam Hodgkinson.

Gesine Reinert -- University of Oxford

Differential Stein operators for multivariate continuous distributions and applications

Abstract: The Stein operator approach has been fruitful for one-dimensional distributions. In this paper we generalise it to multivariate continuous distributions. We shall discuss different notions for the Stein kernel. Among the applications we consider is the Wasserstein distance between two continuous probability distributions under the assumption of existence of a Poincaré constant.

This is joint work with Guillaume Mijoule and Yvik Swan (Liege).

Adrian Röllin -- National University of Singapore

Rates for the normal approximation of the triangle count in the Erdös-Rényi random graph

Abstract: In 1989, Barbour, Karonski and Rucinski obtained error bounds for the normal approximation of subgraph counts in the classic Erdös-Rényi random graph. Their bounds were given in Wasserstein metric and are arguably best possible. Almost 30 years later, we can finally complement their result by matching bounds in Kolmogorov metric for the special case of triangles. The proofs are based on a new variant of the Stein-Tikhomirov method — a combination of Stein’s method and characteristic functions introduced by Tikhomirov (1980).

Dominic Schuhmacher -- Georg-August-Universität Göttingen

Convergence rates for geostatistical thinning of point processes based on Gaussian random fields

Abstract: In a random thinning of a point process $\Xi$ on $\mathbb{R}^d$, survival or extinction is decided by tossing (possibly asymmetric and dependent) coins for each point. It is often convenient to model a thinning by a $[0,1]$-valued random field $\pi$ of survival probabilities on $\mathbb{R}^d$ and to assume that, given $\Xi$ and $\pi$, a point at $x$ survives with probability $\pi(x)$ independently of any other survivals.

It is well known that, under conditions, an increasingly dense process thinned by an asymptotically vanishing random field converges in distribution towards a Cox or even Poisson process. In this talk we consider the stationary situation and assume that $\pi$ can be expressed as a $[0,1]$-valued function of a Gaussian random field that is independent of $\Xi$. We present rates in Wasserstein distance for approximation of the thinning by a Poisson process, which depend in intuitive ways on the decay of the correlation function of the Gaussian random field, its behaviour at $0$ and the orderliness of the point process. We use various tools from Stein's method for the proof.

Qi-Man Shao -- Chinese University of Hong Kong

Cramér Type Moderate Deviations: Self-normalization vs Deterministic Normalization

Abstract: The Cramér type moderate deviation quantifies accuracy of the relative error of distributional approximation and can provide a theoretical justification for the use of limiting tail probability. In this talk we shall review recent progress on Cramér type moderate deviation  for self-normalized sums of independent random variables, self-normalized martingales and self-normalized quantile estimator. It shows that the moment assumptions under self-normalization are much weaker than those under deterministic normalization.

Simon Tavaré -- Cancer Research UK Cambridge Institute

Spaghetti loops and a playground game: simulating the component counts of combinatorial structures

Abstract: We describe three methods for simulating the cycle counts of random permutations and compare their efficiencies. The Feller Coupling method simulates a permutation of size $n$ with $O(\log{n})$ calls to a random number generator. We show how to exploit the Feller coupling to provide a very fast method for simulating logarithmic assemblies, such as random mappings, more generally.  We illustrate by estimating the probability that a random mapping has no repeated component sizes. For logarithmic multisets and selections, this approach is replaced by an acceptance/rejection method based on a particular conditioning relationship that represents the distribution of the combinatorial structure as that of independent random variables conditioned on a weighted sum, and we show how to improve its acceptance rate. This is joint work with Richard Arratia, Andrew Barbour and Warren Ewens. Our work was motivated by studying the `Ends of Spaghetti Formula’, which gives the distribution of the loop lengths when the ends of spaghetti strands are tied together at random. I will conclude the talk with a discussion of a children’s game that makes components of size 1 impossible.

Sergey Utev -- University of Leicester

Open probability problems

Abstract: TBA

Volodymyr Vaskovych -- La Trobe University

Rate of convergence of functionals of Gaussian random fields with long-range dependence

Abstract: Depending on the values of parameters, local functionals of homogeneous Gaussian random fields with long range dependence can converge in distribution to either Gaussian or non-Gaussian Hermite-type random variables. We investigate Lévy concentration function of Hermite-type random variables and use the obtained result to derive the rate of convergence of these functionals in the non-central limit case. To the best of our knowledge, this is the first result in the literature on the rates of convergence to Hermite-type distributions with ranks greater than 2. The results were obtained under rather general assumptions on the spectral densities of the random fields.

Yuting Wen -- The University of Melbourne

Stein's method for zero-inflated Poisson approximations

Abstract: A zero-inflated Poisson distribution is a mixture of Poisson distribution and a Dirac measure at 0. It has been used extensively in statistical modelling for the counts of rare events in random environments, but there are few theoretical studies to justify the use of this model. This paper aims to close the gap. It is well-known that for dependent sums, a natural tool is Stein's method, which has so far been used for uni-modal distribution approximation. Since zero-inflated Poisson distributions are multi-modal, there are several difficulties to overcome to achieve a tight bound. This is a joint work with Aihua Xia.

Jin-Ting Zhang -- National University of Singapore

A Simple Scale-Invariant Two-Sample Test for High-Dimensional Data

Abstract: Recently, several non-scale and scale-invariant tests have been proposed for two-sample problems for high-dimensional data. Most of them impose strong assumptions on the underlying covariance matrix so that their test statistics are asymptotically normally distributed. However, in practice, these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well. In this paper, we propose a simple scale-invariant two-sample test which has  good size control and power without imposing strong assumptions on the underlying covariance or correlation matrix.  A simulation study and a real data example  demonstrate the good performance of the proposed test, via comparing it against several well-known non-scale  and scale-invariant tests. Joint work with Liang Zhang and Tianming Zhu.

Xiaowen Zhou -- Concordia University

A continuous-state nonlinear branching process

Abstract: A continuous-state branching process can be  identified as the unique nonnegative solution to a SDE driven by a Brownian motion and a compensated Poisson random measure; see for example, Dawson and Li (2012). By adapting this SDE, we can introduce a continuous-state branching process with nonlinear branching mechanism. Intuitively, the  solution to the modified SDE is a branching process  with branching rates depending on the current population size.

Using a martingale approach, we  study its survival/extinction behaviors and find respective sufficient conditions on the branching parameters under which the process either survives with probability one or dies out with a positive probability. Similarly, we can also discuss the explosion behaviors for the continuous-state nonlinear branching process. We will show that those conditions are quite sharp.

This talk is based  on joint work with Peisen li and Xu Yang.