The Optimal Sink and the Best Source in a Markov Chain

@article{Bakhtin2011TheOS,
  title={The Optimal Sink and the Best Source in a Markov Chain},
  author={Yuri Bakhtin and Leonid A. Bunimovich},
  journal={Journal of Statistical Physics},
  year={2011},
  volume={143},
  pages={943-954}
}
It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a… 
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References

SHOWING 1-10 OF 16 REFERENCES
Hitting and return times in ergodic dynamical systems
Given an ergodic dynamical system (X, T,μ), and U C X measurable with μ(U) > 0, let μ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with μ(U n ) →
Where to place a hole to achieve a maximal escape rate
A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very
ASYMPTOTICS FOR HITTING TIMES
(essentially decreasing sequences of balls in ametric space X) and identified to be the distribution function of the posi-tive exponential law with parameter 1, in many classes of mixing systems,in
The limiting distribution and error terms for return times of dynamical systems
We develop a new framework that allows to prove that the limiting distribution of return times for a large class of mixing dynamical systems are Poisson distributed. We demonstrate our technique
Possible limit laws for entrance times of an ergodic aperiodic dynamical system
LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ0∞G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and
Sharp error terms and neccessary conditions for exponential hitting times in mixing processes
We prove an upper bound for the error in the exponential approximation of the hitting time law of a rare event in α-mixing processes with exponential decay, ϕ-mixing processes with a summable
On the notion of recurrence in discrete stochastic processes
Received by the editors March 3, 1947. 1 John Simon Guggenheim Memorial Fellow. 2 That the theory of stationary stochastic processes is mathematically equivalent with an "ergodic" theory (to which
Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources
TLDR
This work provides a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources and proves two strong approximation results from which it can be deduced pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum.
Which hole is leaking the most: a topological approach to study open systems
We study the process of escape of orbits through a hole in the phase space of a dynamical system generated by a chaotic map of an interval. If this hole is an element of Markov partition we are able
An Introduction To Probability Theory And Its Applications
TLDR
A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
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