New publications

Spectral norm bounds for block Markov chain random matrices

We have submitted Spectral norm bounds for block Markov chain random matrices, and it is currently under review. This is joint work between Albert Senen-Cerda and myself. A preprint is available on arXiv.


This paper quantifies the asymptotic order of the largest singular value of a centered random matrix built from the path of a Block Markov Chain (BMC). In a BMC there are n labeled states, each state is associated to one of K clusters, and the probability of a jump depends only on the clusters of the origin and destination. Given a path X0, X1, …, XTn started from equilibrium, we construct a random matrix N that records the number of transitions between each pair of states. We prove that if ω(n)=Tn=o(n*n), then ∥N−E[N]∥=ΩP(sqrt(Tn/n)). We also prove that if Tn=Ω(nlnn), then ∥N−E[N]∥=OP(sqrt(Tn/n)) as n→∞; and if Tn=ω(n), a sparser regime, then ∥NΓ−E[N]∥=OP(sqrt(Tn/n)). Here, NΓ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is ΘP(sqrt(Tn/n)) for BMCs.


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Jaron Sanders received in 2012 M.Sc. degrees in Mathematics and Physics from the Eindhoven University of Technology, The Netherlands, as well as a PhD degree in Mathematics in 2016. After he obtained his PhD degree, he worked as a post-doctoral researcher at the KTH Royal Institute of Technology in Stockholm, Sweden. Jaron Sanders then worked as an assistant professor at the Delft University of Technology, and now works as an assistant professor at the Eindhoven University of Technology. His research interests are applied probability, queueing theory, stochastic optimization, stochastic networks, wireless networks, and interacting (particle) systems.