Markov chains with heavy-tailed increments and asymptotically zero drift
(2019)
Journal Article
Georgiou, N., Menshikov, M. V., Petritis, D., & Wade, A. R. (2019). Markov chains with heavy-tailed increments and asymptotically zero drift. Electronic Journal of Probability, 24, Article 62. https://doi.org/10.1214/19-ejp322
Outputs (48)
Localisation in a growth model with interaction (2018)
Journal Article
Costa, M., Menshikov, M., Shcherbakov, V., & Vachkovskaia, M. (2018). Localisation in a growth model with interaction. Journal of Statistical Physics, 171(6), 1150-1175. https://doi.org/10.1007/s10955-018-2055-4This paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite cycle graph. The model can be regarded as a reinforced urn model with graph-based interaction. It is motivated by coopera... Read More about Localisation in a growth model with interaction.
Long term behaviour of two interacting birth-and-death processes (2018)
Journal Article
Menshikov, M., & Shcherbakov, V. (2018). Long term behaviour of two interacting birth-and-death processes. Markov processes and related fields, 24(1), 85-106In this paper we study the long term evolution of a continuous time Markov chain formed by two interacting birth-and-death processes. The interaction between the processes is modelled by transition rates that are given by functions with suitable mono... Read More about Long term behaviour of two interacting birth-and-death processes.
Heavy-tailed random walks on complexes of half-lines (2017)
Journal Article
Menshikov, M. V., Petritis, D., & Wade, A. R. (2018). Heavy-tailed random walks on complexes of half-lines. Journal of Theoretical Probability, 31(3), 1819-1859. https://doi.org/10.1007/s10959-017-0753-5We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is gove... Read More about Heavy-tailed random walks on complexes of half-lines.
Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems (2016)
Book
Menshikov, M., Popov, S., & Wade, A. (2016). Non-Homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press. https://doi.org/10.1017/9781139208468
Anomalous recurrence properties of many-dimensional zero-drift random walks (2016)
Journal Article
Georgiou, N., Menshikov, M. V., Mijatovic, A., & Wade, A. R. (2016). Anomalous recurrence properties of many-dimensional zero-drift random walks. Advances in Applied Probability, 48(Issue A), 99-118. https://doi.org/10.1017/apr.2016.44Famously, a d-dimensional, spatially homogeneous random walk whose increments are nondegenerate, have finite second moments, and have zero mean is recurrent if d∈{1,2}, but transient if d≥3. Once spatial homogeneity is relaxed, this is no longer true... Read More about Anomalous recurrence properties of many-dimensional zero-drift random walks.
Random dynamical systems with systematic drift competing with heavy-tailed randomness (2016)
Journal Article
Belitsky, V., Menshikov, M., Petritis, D., & Vachkovskaia, M. (2016). Random dynamical systems with systematic drift competing with heavy-tailed randomness. Markov processes and related fields, 22(4), 629-652Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology --- we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems i... Read More about Random dynamical systems with systematic drift competing with heavy-tailed randomness.
Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach (2014)
Journal Article
Menshikov, M., & Petritis, D. (2014). Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach. Stochastic Processes and their Applications, 124(7), 2388-2414. https://doi.org/10.1016/j.spa.2014.03.001We establish general theorems quantifying the notion of recurrence–through an estimation of the moments of passage times–for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomeno... Read More about Explosion, implosion, and moments of passage times for continuous-time Markov chains: A semimartingale approach.
Random walk in mixed random environment without uniform ellipticity (2013)
Journal Article
Hryniv, O., Menshikov, M. V., & Wade, A. R. (2013). Random walk in mixed random environment without uniform ellipticity. Proceedings of the Steklov Institute of Mathematics, 282(1), 106-123. https://doi.org/10.1134/s0081543813060102We study a random walk in random environment on ℤ+. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities d... Read More about Random walk in mixed random environment without uniform ellipticity.
Excursions and path functionals for stochastic processes with asymptotically zero drifts (2013)
Journal Article
Hryniv, O., Menshikov, M. V., & Wade, A. R. (2013). Excursions and path functionals for stochastic processes with asymptotically zero drifts. Stochastic Processes and their Applications, 123(6), 1891-1921. https://doi.org/10.1016/j.spa.2013.02.001We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not... Read More about Excursions and path functionals for stochastic processes with asymptotically zero drifts.