The Distribution of Mixing times in Markov Chains
نویسنده
چکیده
The distribution of the “mixing time” or the “time to stationarity” in a discrete time irreducible Markov chain, starting in state i, can be defined as the number of trials to reach a state sampled from the stationary distribution of the Markov chain. Expressions for the probability generating function, and hence the probability distribution of the mixing time starting in state i are derived and special cases explored. This extends the results of the author regarding the expected time to mixing [J.J. Hunter, Mixing times with applications to perturbed Markov chains, Linear Algebra Appl. 417 (2006) 108–123], and the variance of the times to mixing, [J.J. Hunter, Variances of first passage times in a Markov chain with applications to mixing times, Linear Algebra Appl. 429 (2008) 1135–1162]. Some new results for the distribution of recurrence and first passage times in three-state Markov chain are also presented. MSC2010 classification: 37A25, 60J10, 60E05, 60J22
منابع مشابه
Random generation of 2 times 2 times ... times 2 times J contingency tables
We propose two Markov chains for sampling (m + 1)-dimensional contingency tables indexed by {1, 2}m × {1, 2, . . . , n}. Stationary distributions of our chains are the uniform distribution and a conditional multinomial distribution (which is equivalent to the hypergeometric distribution ifm=1). Mixing times of our chains are bounded by ( 1 2 )n(n− 1) ln(N/(2m ))= ( 1 2 )n(n− 1) ln(dn/ ), where ...
متن کاملRelaxation of Product Markov Chains on Product Spaces
The purpose of the paper is studying the relaxation time of product{ type Markov chains on product spaces which approach a product distribution. We determine bounds to approach stationarity for such Markov chains in terms of the mixing times of the component Markov chains. In cases where the component mixing times vary much we propose an optimized visiting scheme which makes such product{type M...
متن کاملMixing times of Lozenge Tiling and Card Shuffling Markov Chains
We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall, and Sinclair to generate random tilings of regions by lozenges. For an l×l region we bound the mix...
متن کاملMarkov chains and mixing times ( part 2 - coupling )
This week’s post continues last week’s discussion of Markov chains and mixing times, and introduces the idea of coupling as a method for estimating mixing times. We remark that some nice notes on the subject of coupling (and others) can be found on Steve Lalley’s web page – of particular relevance for our purposes here are the notes on “Convergence rates of Markov chains”. A more thorough and c...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- APJOR
دوره 30 شماره
صفحات -
تاریخ انتشار 2013