A/Prof Kais Hamza

Research Overview

Random Walks with Long Memory: A. Collevecchio (AC) and I introduced a new type of multi-dimensional bootstrap random walks with very interesting and unexpected properties. Investigations of so-called General Bootstrap Random Walks (with AC & R Williams) and of site percolation on a "bootstrap" lattice (with AC & A Holroyd) are in progress. AC, L Tournier and I studied the behaviour of a deterministic walk on the randomly oriented Manhattan lattice and have shown that the process localizes on two vertices.

Mimicking Self-similar Processes: JY Fan (JYF), F Klebaner (FK) and I constructed non-standard martingales with given marginals. This area is of significant mathematical research activity with applications to the theory of pricing in financial markets.

Mathematical Biology: P Jagers (PJ), FK and I extended the classical branching setting in a model of complex populations using measure-valued processes We established a Law of Large Numbers for population-dependent branching processes, a Central Limit Theorem (with JYF) and extended the results to a multi-type setting.

Selected Publications

[1] Fan, Jie Yen ; Hamza, Kais ; Jagers, Peter ; Klebaner, Fima. "Convergence of the age structure of general schemes of population processes". In: Bernoulli. 2020 ; Vol. 26, No. 2. pp. 893-926. https://doi.org/10.3150/18-BEJ1100

[2] Collevecchio, Andrea ; Hamza, Kais ; Tournier, Laurent. "A deterministic walk on the randomly oriented Manhattan lattice". In: Electronic Journal of Probability. 2019 ; Vol. 24. https://doi.org/10.1214/19-EJP385

[3] Baker, Jeremy ; Chigansky, P. ; Hamza, K. ; Klebaner, F. C. "Persistence of small noise and random initial conditions". In: Advances in Applied Probability. 2018 ; Vol. 50, No. A. pp. 67-81. https://doi.org/10.1017/apr.2018.71

[4] Collevecchio, Andrea ; Hamza, Kais ; Shi, Meng. "Bootstrap random walks". In: Stochastic Processes and their Applications. 2016 ; Vol. 126, No. 6. pp. 1744-1760. https://doi.org/10.1016/j.spa.2015.11.016

[5] Fan, Jie Yen ; Hamza, Kais ; Klebaner, Fima C. "Mimicking self-similar processes". In: Bernoulli. 2015 ; Vol. 21, No. 3. pp. 1341-1360. https://doi.org/10.3150/13-BEJ588