Discovery Projects  Grant ID: DP150101459
 Title
 Discovery Projects  Grant ID: DP150101459
 Funding
 ARC  Discovery Projects
 Contract (GA) number
 DP150101459
 Start Date
 2015/01/01
 End Date
 2017/12/31
 Open Access mandate
 no
 Organizations
 
 More information
 http://purl.org/auresearch/grants/arc/DP150101459

A dichotomy for CLT in total variation
Let $\eta_i$, $i\ge 1$, be a sequence of independent and identically distributed random variables with finite third moment, and let $\Delta_n$ be the total variation distance between the distribution of $S_n:=\sum_{i=1}^n\eta_i$ and the normal distribution with the same mean and variance. In this note, we show the dichotomy that either $\Delta_n=1$ for all $n$ or $\Delta_n=O\left(n^{1/2}\right)$.P\'olya urns with immigration at random times
Peköz, Erol; Röllin, Adrian; Ross, Nathan (2016)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We study the number of white balls in a classical P\'olya urn model with the additional feature that, at random times, a black ball is added to the urn. The number of draws between these random times are i.i.d. and, under certain moment conditions on the interarrival distribution, we characterize the limiting distribution of the (properly scaled) number of white balls as the number of draws goes to infinity. The possible limiting distributions obtained in this way vary considerably depending...A marked renewal process model for the size of a honey bee colony
Xia, Aihua; Huggins, Richard M.; Barons, Martine J.; Guillot, Louis (2016)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)Many areas of agriculture rely on honey bees to provide pollination services and any decline in honey bee numbers can impact on global food security. In order to understand the dynamics of honey bee colonies we present a discrete time marked renewal process model for the size of a colony. We demonstrate that under mild conditions this attains a stationary distribution that depends on the distribution of the numbers of eggs per batch, the probability an egg hatches and the distributions of the...Wireless network signals with moderately correlated shadowing still appear Poisson
Ross, Nathan; Schuhmacher, Dominic (2016)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We consider the point process of signal strengths emitted from transmitters in a wireless network and observed at a fixed position. In our model, transmitters are placed deterministically or randomly according to a hard core or Poisson point process and signals are subjected to power law path loss and random propagation effects that may be correlated between transmitters. We provide bounds on the distance between the point process of signal strengths and a Poisson process with the same mean m...Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
Staudte, Robert; Xia, Aihua (2017)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We demonstrate that questions of convergence and divergence regarding shapes of distributions can be carried out in a location and scalefree environment. This environment is the class of probability density quantiles (pdQs), obtained by normalizing the composition of the density with the associated quantile function. It has earlier been shown that the pdQ is representative of a locationscale family and carries essential information regarding shape and tail behavior of the family. The class...The Probability of Intransitivity in Dice and Close Elections
Hązła, Jan; Mossel, Elchanan; Ross, Nathan (2018)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)The phenomenon of intransitivity in elections, where the pairwise orderings of three or more candidates induced by voter preference is not transitive, was first observed by Condorcet in the 18th century, and is fundamental to modern social choice theory. There has been some recent interest in understanding intransitivity for three or more $n$sided dice (with nonstandard labelings), where now the pairwise ordering is induced by the probability, relative to $1/2$, that a throw from one dice i...Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs
Reinert, Gesine; Ross, Nathan (2017)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We provide a general bound on the Wasserstein distance between two arbitrary distributions of sequences of Bernoulli random variables. The bound is in terms of a mixing quantity for the Glauber dynamics of one of the sequences, and a simple expectation of the other. The result is applied to estimate, with explicit error, expectations of functions of random vectors for some Ising models and exponential random graphs in "high temperature" regimes.Dirichlet approximation of equilibrium distributions in Cannings models with mutation
Gan, H. L.; Röllin, Adrian; Ross, Nathan (2016)
Projects: ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)Consider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite population stationary distribution for the WrightFisher gen...Error bounds in local limit theorems using Stein's method
Barbour, A. D.; Röllin, Adrian; Ross, Nathan (2017)
Projects: ARC  Discovery Projects  Grant ID: DP150103588 (DP150103588), ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erd\H{o}sR\'enyi random graph, and of the CurieWeiss model of magnetism, where we provide optimal or near optimal rate...Approximations and Limit Theorems for DiscreteTime Occupancy Processes
Hodgkinson, Liam; McVinish, Ross; Pollett, Philip K. (2018)
Projects: ARC  ARC Centres of Excellences  Grant ID: CE140100049 (CE140100049), ARC  Discovery Projects  Grant ID: DP150101459 (DP150101459)We study deterministic and normal approximations for a class of discretetime occupancy processes, namely, Markov chains with transition kernels of product Bernoulli form. This class encompasses numerous models which appear in the complex networks literature, including stochastic patch occupancy models in ecology, network models in epidemiology, and a variety of dynamic random graph models. Moment inequalities on the deviation from an analogous deterministic model are presented, alongside bou... 
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