2015 Melbourne-Singapore Probability and Statistics Forum



Wednesday, 25 February, 2015


University of Melbourne
Richard Berry Building
Evan Williams Theatre (G03) (map)

Speakers and talks:


Louis H. Y. Chen
National University of Singapore
On the error bound in the normal approximation for Jack measures

The one-parameter family of Jack_α measures on partitions of n is an important discrete analog of Dyson's β ensembles of random matrix theory. Except for α=1/2, 1, 2, which have group theoretic interpretations, the Jack_α measure is difficult to analyze. In the case α=1, the Jack measure agrees with the Plancherel measure on the irreducible representations of the symmetric group S_n, parametrized by the partitions of n. The normal approximation for the character ratio evaluated at the transposition (12) under the Plancherel measure has been well studied, notably by Fulman (2005, 2006) and Shao and Su (2006). A generalization of the character ratio under the Jack_α measure has also been studied by Fulman (2004, 2006) and Fulman and Goldstein (2011). In this talk, we present results on both uniform and non-uniform error bounds on the normal approximation for the Jack_α measure for α>0. Our results improve those in the literature and come close to a conjecture of Fulman (2004). Our proofs use Stein's method and zero-bias coupling. This talk is based on joint work with Le Van Thanh.

Aurore Delaigle
The University of Melbourne
Deconvolution when the Error Distribution is Unknown

In nonparametric deconvolution problems, in order to estimate consistently a density or distribution from a sample of data contaminated by additive random noise it is often assumed that the noise distribution is completely known or that an additional sample of replicated or validation data is available. Methods have also been suggested for estimating the scale of the error distribution, but they require somewhat restrictive smoothness assumptions on the signal distribution, which can be hard to verify in practice. Taking a completely new approach to the problem, we argue that data rarely come from a simple, regular distribution, and that this can be exploited to estimate the signal distributions using a simple procedure, often giving very good performance. Our method can be extended to other problems involving errors-in-variables, such as nonparametric regression estimation. Its performance in practice is remarkably good, often equalling (even unexpectedly) the performance of techniques that use additional data to estimate the unknown error distribution. This is joint work with Peter Hall.

Adrian Röllin
National University of Singapore
Using Stein's method to calculate variances

Stein's method is best known as a very efficient tool for distributional approximation. When used for normal approximation, the bounds are typically expressed in terms of the ratio between the third or maybe higher moments of the involved random variables and the variance of the quantity of interest. Whereas finding upper bounds on the higher moments is often not too difficult, calculating the variance or finding lower bounds on it can be challenging. In this talk, we show how Stein's method can be used to calculate the asymptotic behaviour of the variance in various problems. These asymptotics are expressed in terms of functionals of limiting probabilistic objects that arise from couplings used in Stein's method.

Shaun McKinlay
The University of Melbourne
From Levy's arcsine laws to iterated random functions

In an attempt to give a simple probabilistic argument linking two of Levy's arcsine laws for Brownian motion, we arrive at a random equation (RE) with an arcsine distributed solution. Extending a result due to Pitman (1937), we solve the RE by solving an equivalent RE with a gamma distributed solution. Since backward iterations of the RE describe an interval splitting scheme, while forward iterations generate an ergodic Markov chain, the solution is the limiting distribution of these iterations. Generalising the RE in several ways, and armed with our new technique, we prove a number of new results on the limiting distributions of these processes, and generalisations thereof.

Peter Taylor
The University of Melbourne
A model for the BitCoin block chain that takes propagation delays into account

Unlike cash transactions, most electronic transactions require the presence of a trusted authority to verify that the payer has sufficient funding to be able to make the transaction and to adjust the account balances of the payer and payee.

In recent years 'BitCoin' has been proposed as an 'electronic equivalent of cash'. The general idea is that transactions are verified in a coded form in a 'block chain', which is maintained by the community of participants.

Problems can arise when the block chain splits: that is, different participants have different versions of the block chain, something which can happen only when there are propagation delays, at least if all participants are behaving according to the protocol. In this talk I shall present some models for analysing different aspects of the splitting behaviour of the block chain. I shall then go on to perform a similar analysis for a situation where a group of participants has adopted the recently-proposed 'selfish mine' strategy for gaining a greater advantage from BitCoin processing than its combined computer power should be able to generate.

Joint work with Johannes Göbel, Paul Keeler and Tony Krzesinski.



    10:00am — 10:50am     Chen  
    11:00am — 11:50am     Delaigle  
    12:00pm — 1:30pm     Lunch  
    1:30pm — 2:20pm     Röllin  
    2:30pm — 3:20pm     Mckinlay  
    3:30pm — 4:00pm     Break  
    4:00pm — 4:50pm     Taylor  
    6:00pm     Dinner*  
  *RSVP to Nathan Ross by 23 Feb.