arXiv subjects

Thomas Kaijser

Publications and source records attributed to Thomas Kaijser.

4 recordsLinked to original sources

arXiv2016

Stochastic perturbations of iterations of a simple, non-expanding, nonperiodic, piecewise linear, interval-map

Let g(x)=x/2 + 17/30 (mod 1), let ξ_i, i= 1,2,... be a sequence of independent, identically distributed random variables with uniform distribution on the interval [0,1/15], define g_i(x)=g(x)+ ξ_i (mod 1) and, for n=1,2,..., define g^n(x)=g_n(g_{n-1}(...(g_1(x))...)). For x \in [0,1) let μ_{n,x} denote the distribution of g^n(x). The purpose of this note is to show that there exists a unique probability measure μ, such that, for all x \in [0,1), μ_{n,x} tends to μ, as n tends to infinity. This contradicts a claim by Lasota and Mackey from 1987 stating that the process has an asymptotic three-periodicity.

math.PR↗

arXiv2013

Convergence in distribution for filtering processes associated to Hidden Markov Models with densities

Consider a filtering process associated to a hidden Markov model with densities for which both the state space and the observation space are complete, separable, metric spaces. If the underlying, hidden Markov chain is strongly ergodic and the filtering process fulfills a certain coupling condition we prove that, in the limit, the distribution of the filtering process is independent of the initial distribution of the hidden Markov chain. If furthermore the hidden Markov chain is uniformly ergodic, then we prove that the filtering process converges in distribution.

math.PR↗

arXiv2010

On Markov chains induced by partitioned transition probability matrices

Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P. Let K denote the set of probability vectors on S. To every partition M of P we can associate a transition probability function on K defined in such a way that if p in K and m in M are such that ||pm|| > 0, then, with probability ||pm|| the vector p is transferred to the vector pm/||pm||. Here ||.|| denotes the l_1-norm. In this paper we investigate convergence in distribution for Markov chains generated by transition probability functions induced by partitions of transition probability matrices. An important application of the convergence results obtained is to filtering processes of partially observed Markov chains.

math.PR↗

arXiv2009

Improving the primal-dual algorithm for the transportation problem in the plane

The transportation problem in the plane - how to move a set of objects from one set of points to another set of points in the cheapest way - is a very old problem going back several hundreds of years. In recent years the solution of the problem has found applications in the analysis of digital images when searching for similarities and discrepancies between images. The main drawback, however, is the long computation time for finding the solution. In this paper we present some new results by which the time for solving the transportation problem in the plane can be reduced substantially. As cost-function we choose a distance-function between points in the plane. We consider both the case when the distance-function is equal to the ordinary Euclidean distance, as well as the case when the distance-function is equal to the square of the Euclidean distance. This latter distance-function has the advantage that it is integer-valued if the coordinates of the points in the plane are integers.

math.OC↗