The detection of words and an ordering for Markov chains

2001 ◽  
Vol 38 (A) ◽  
pp. 66-77 ◽  
Author(s):  
Albrecht Irle ◽  
Joseph Gani
Keyword(s):  
Markov Chains ◽  
Specific Pattern ◽  
Finite Alphabet ◽  
Finite State Automaton ◽  
Stochastic Monotonicity ◽  
Level Crossing ◽  
Stochastic Automata ◽  
Finite State ◽  
First Time ◽  
Random Times

This paper considers the occurrence of patterns in sequences of independent trials from a finite alphabet; Gani and Irle (1999) have described a finite state automaton which identifies exactly those sequences of symbols containing the specific pattern, which may be thought of as the word of interest. Each word generates a particular Markov chain. Motivated by a result of Guibas and Odlyzko (1981) on stochastic monotonicity for the random times when a particular word is completed for the first time, a new level-crossing ordering is introduced for stochastic processes. A process {Yn : n = 0, 1, …} is slower in level-crossing than a process {Zn}, if it takes {Yn} stochastically longer than {Zn} to exceed any given level. This relation is shown to be useful for the comparison of stochastic automata, and is used to investigate this ordering for Markov chains in discrete time.

The detection of words and an ordering for Markov chains

2001 ◽  
Vol 38 (A) ◽  
pp. 66-77
Author(s):  
Albrecht Irle ◽  
Joseph Gani
Keyword(s):  
Markov Chains ◽  
Specific Pattern ◽  
Finite Alphabet ◽  
Finite State Automaton ◽  
Stochastic Monotonicity ◽  
Level Crossing ◽  
Stochastic Automata ◽  
Finite State ◽  
First Time ◽  
Random Times

This paper considers the occurrence of patterns in sequences of independent trials from a finite alphabet; Gani and Irle (1999) have described a finite state automaton which identifies exactly those sequences of symbols containing the specific pattern, which may be thought of as the word of interest. Each word generates a particular Markov chain. Motivated by a result of Guibas and Odlyzko (1981) on stochastic monotonicity for the random times when a particular word is completed for the first time, a new level-crossing ordering is introduced for stochastic processes. A process {Yn : n = 0, 1, …} is slower in level-crossing than a process {Zn }, if it takes {Yn } stochastically longer than {Zn } to exceed any given level. This relation is shown to be useful for the comparison of stochastic automata, and is used to investigate this ordering for Markov chains in discrete time.

scholarly journals Combable functions, quasimorphisms, and the central limit theorem

2009 ◽  
Vol 30 (5) ◽  
pp. 1343-1369 ◽  
Author(s):  
DANNY CALEGARI ◽  
KOJI FUJIWARA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Word Length ◽  
Hyperbolic Groups ◽  
Finite State Automaton ◽  
List Type ◽  
Generating Sets ◽  
Finite State ◽  
Word Hyperbolic Groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

An extension of a convergence theorem for Markov chains arising in population genetics

2016 ◽  
Vol 53 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Martin Möhle ◽  
Morihiro Notohara
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Convergence Theorem ◽  
Transition Matrix ◽  
Initial Distribution ◽  
Generator Matrix ◽  
Finite Dimensional ◽  
Finite State ◽  
Initial Distributions ◽  
Finite State Space

AbstractAn extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

scholarly journals Survey-based Impact of Influencing Parameters on Level Crossings Safety

2016 ◽  
Vol 28 (6) ◽  
pp. 639-649 ◽  
Author(s):  
Martin Starčević ◽  
Danijela Barić ◽  
Hrvoje Pilko
Keyword(s):  
Survey Questionnaire ◽  
Level Crossing ◽  
Level Crossings ◽  
Policy Makers ◽  
Railway Traffic ◽  
Research Activities ◽  
On Line ◽  
Line Survey ◽  
First Time ◽  
Influential Parameters

Level crossing (LC) accidents are a significant safety challenge worldwide and for that reason they have been subject of numerous research activities. Joint conclusion is that human behaviour is the main cause of accidents. This study examines how and to which extent certain influential parameters cause accident mechanisms on level crossings. To gain the necessary data we used an on-line survey questionnaire that was sent internationally to key experts in the field of road and railway safety. A total of 185 experts were asked to rank how much certain parameters influence level crossings accident mechanisms and what are the best countermeasures for diminishing accidents at level crossings. To our knowledge, this is the first time that an international survey among key experts was used to gain necessary data about influential parameters regarding level crossings safety. The results of this study could be used by road and railway traffic engineers and policy makers for further enhancement of LC’s safety.

Sign in / Sign up

Export Citation Format

Share Document