arXiv subjects

Ryszard Szekli

Publications and source records attributed to Ryszard Szekli.

5 recordsLinked to original sources

arXiv2019

Some remarks on associated random fields, random measures and point processes

In this paper, we first show that for a countable family of random elements taking values in a partially ordered Polish space (POP), association (both positive and negative) of all finite dimensional marginals implies that of the infinite sequence. Our proof proceeds via Strassen's theorem for stochastic domination and thus avoids the assumption of normally ordered on the product space as needed for positive association in [Lindqvist 1988]. We use these results to show on Polish spaces that finite dimensional negative association implies negative association of the random measure and negative association is preserved under weak convergence of random measures. The former provides a simpler proof in the most general setting of Polish spaces complementing the recent proofs in [Poinas et al. 2017] and [Lyons 2014] which restrict to point processes in Euclidean spaces and locally compact Polish spaces respectively. We also provide some examples of associated random measures which shall illustrate our results as well.

math.PR↗

arXiv2014

Strong Stationary Duality for Möbius monotone Markov chains: examples

We construct strong stationary dual chains for Ising model on a circle, non-symmetric random walk on square lattice and a random walk on hypercube. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to natural orderings of the corresponding state spaces. This method provides an easy way to find eigenvalues in the Ising model and for a random walk on hypercube. Keywords: Markov chains; stochastic monotonicity; eigenvalues; Möbius monotonicity; strong stationary duality; strong stationary times; separation distance; mixing time; Ising model; hypercube

math.PR↗

arXiv2013

Computable bounds on the spectral gap for unreliable Jackson networks

The goal of this paper is to identify exponential convergence rates and to find computable bounds for them for Markov processes representing unreliable Jackson networks. First we use the bounds of Lawler and Sokal in order to show that, for unreliable Jackson networks, the spectral gap is strictly positive if and only if the spectral gaps for the corresponding coordinate birth and death processes are positive. Next, utilizing some results on birth and death processes, we find bounds on the spectral gap for network processes in terms of the hazard and equilibrium functions of the one dimensional marginal distributions of the stationary distribution of the network. These distributions must be in this case strongly light-tailed, in the sense that their discrete hazard functions have to be separated from zero. We relate these hazard functions with the corresponding networks' service rate functions using the equilibrium rates of the stationary one dimensional marginal distributions. We compare the obtained bounds on the spectral gap with some other known bounds.

math.PR↗

arXiv2011

Strong Stationary Duality for Möbius Monotone Markov Chains: Unreliable Networks

For Markov chains with a partially ordered finite state space we show strong stationary duality under the condition of Möbius monotonicity of the chain. We show relations of Möbius monotonicity to other definitions of monotone chains. We give examples of dual chains in this context which have transitions only upwards. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an application to networks of queues.

math.PR↗

arXiv2010

Comparisons and asymptotics for empty space hazard functions of germ-grain models

We study stochastic properties of the empty space for stationary germ-grain models in $\R^d$, in particular we deal with the inner radius of the empty space with respect to a general structuring element which is allowed to be lower-dimensional. We consider Poisson cluster germ-grain models and Boolean models with grains that are clusters of convex bodies and show that more variable size of clusters results in stochastically greater empty space in terms of the empty space hazard function. We also study impact of clusters being more spread in the space on the value of the empty space hazard. Further we obtain asymptotic behavior of the empty space hazard functions at zero and at infinity.

math.PR↗