Probability Measures on Topological Semigroups

Högnäs, Göran; Mukherjea, Arunava

doi:10.1007/978-1-4757-2388-5_2

Cited by 2 publications

(3 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The explicit representation Y = sup n∈N (U n − (n − 1)) shows that P(Y ≤ y) = n≥0 F (y + n) > 0 if and only if F (y) > 0 and the series n≥0 (1 − F (y + n)) converges. Therefore the system (E, ν) is positive recurrent if and only if the variables U n , n ∈ N, have a finite expectation (for extensions see [22]).…”

Section: Invariant Measuresmentioning

confidence: 99%

See 1 more Smart Citation

Random Dynamical Systems on Ordered Topological Spaces

Kellerer

Winkler

2006

Stoch. Dyn.

View full text Add to dashboard Cite

Let (Xn, n ≥ 0) be a random dynamical system and its state space be endowed with a reasonable topology. Instead of completing the structure as common by some linearity, this study stresses — motivated in particular by economic applications — order aspects. If the underlying random transformations are supposed to be order-preserving, this results in a fairly complete theory. First of all, the classical notions of and familiar criteria for recurrence and transience can be extended from discrete Markov chain theory. The most important fact is provided by the existence and uniqueness of a locally finite-invariant measure for recurrent systems. It allows to derive ergodic theorems as well as to introduce an attract or in a natural way. The classification is completed by distinguishing positive and null recurrence corresponding, respectively, to the case of a finite or infinite invariant measure; equivalently, this amounts to finite or infinite mean passage times. For positive recurrent systems, moreover, strengthened versions of weak convergence as well as generalized laws of large numbers are available.

show abstract

Section: Invariant Measuresmentioning

confidence: 99%

“…Indeed, to give only two typical examples: most work on products of i. i. d. matrices is actually restricted to nonnegative matrices (see e.g. Högnäs/Mukherjea [22,Chapter 4]), while economic processes, modeling dams, insurance risk, queues, storage, traffic etc. (see e.g.…”

mentioning

confidence: 99%

Random Dynamical Systems on Ordered Topological Spaces

Kellerer

Winkler

2006

Stoch. Dyn.

View full text Add to dashboard Cite

show abstract

“…Rescaling π 3 , the inclusion hierarchy yields π 3 = [0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0] π 2 = [0, 4, 6, 2, 0, 1, 0, 2, 3, 4, 0, 1, 2, 0, 2] π 1 = [12, 6, 10, 12, 8, 6] = 2 [6, 3,5,6,4,3] For example, π 2 ({2, 6}), ninth entry from the left, has a weight of 3, since {2, 6} is contained in both R 3 and R 4 . And π 1 = 54π.…”

Section: 4mentioning

confidence: 99%

A hierarchical structure of transformation semigroups with applications to probability limit measures

Budzban¹,

Feinsilver²

2012

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

The structure of transformation semigroups on a finite set is analyzed by introducing a hierarchy of functions mapping subsets to subsets. The resulting hierarchy of semigroups has a corresponding hierarchy of minimal ideals, or kernels. This kernel hierarchy produces a set of tools that provides direct access to computations of interest in probability limit theorems; in particular, finding certain factors of idempotent limit measures. In addition, when considering transformation semigroups that arise naturally from edge colorings of directed graphs, as in the road-coloring problem, the hierarchy produces simple techniques to determine the rank of the kernel and to decide when a given kernel is a right group. In particular, it is shown that all kernels of rank one less than the number of vertices must be right groups and their structure for the case of two generators is described.

show abstract

Probability Measures on Topological Semigroups

Cited by 2 publications

References 57 publications

Random Dynamical Systems on Ordered Topological Spaces

Random Dynamical Systems on Ordered Topological Spaces

A hierarchical structure of transformation semigroups with applications to probability limit measures

Contact Info

Product

Resources

About