arXiv subjects

Lasse Leskelä

Publications and source records attributed to Lasse Leskelä.

45 records · Page 3Linked to original sources

arXiv2011

A local limit theorem for a transient chaotic walk in a frozen environment

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk's probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian.

math.PR↗

arXiv2011

Stochastic order characterization of uniform integrability and tightness

We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. Especially, we show that whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics.

math.PR↗

arXiv2010

A Dilution Test for the Convergence of Subseries of a Monotone Series

Cauchy's condensation test allows to determine the convergence of a monotone series by looking at a weighted subseries that only involves terms of the original series indexed by the powers of two. It is natural to ask whether the converse is also true: Is it possible to determine the convergence of an arbitrary subseries of a monotone series by looking at a suitably weighted version of the original series? In this note we show that the answer is affirmative and introduce a new convergence test particularly designed for this purpose.

math.CA↗

arXiv2010

Stability of a spatial polling system with greedy myopic service

This paper studies a spatial queueing system on a circle, polled at random locations by a myopic server that can only observe customers in a bounded neighborhood. The server operates according to a greedy policy, always serving the nearest customer in its neighborhood, and leaving the system unchanged at polling instants where the neighborhood is empty. This system is modeled as a measure-valued random process, which is shown to be positive recurrent under a natural stability condition that does not depend on the server's scan range. When the interpolling times are light-tailed, the stable system is shown to be geometrically ergodic. The steady-state behavior of the system is briefly discussed using numerical simulations and a heuristic light-traffic approximation.

math.PR↗

arXiv2010

Stability of parallel queueing systems with coupled service rates

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.

math.PR↗

arXiv2009

Computational methods for stochastic relations and Markovian couplings

Order-preserving couplings are elegant tools for obtaining robust estimates of the time-dependent and stationary distributions of Markov processes that are too complex to be analyzed exactly. The starting point of this paper is to study stochastic relations, which may be viewed as natural generalizations of stochastic orders. This generalization is motivated by the observation that for the stochastic ordering of two Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are an algorithmic characterization of stochastic relations between finite spaces, and a truncation approach for comparing infinite-state Markov processes. The methods are illustrated with applications to loss networks and parallel queues.

math.PR↗

arXiv2008

Stochastic relations of random variables and processes

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.

math.PR↗

arXiv2007

Stabilization of an overloaded queueing network using measurement-based admission control

Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks the admission decisions are often based on imperfect measurements on the network state. This paper studies how the lack of complete state information affects the system performance by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters.

math.PR↗

arXiv2007

Scaling limits for random fields with long-range dependence

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $λ$ of the sets grows to infinity and the mean volume $ρ$ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which $λ$ and $ρ$ are scaled. If $λ$ grows much faster than $ρ$ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.

math.PR↗