The Markov Chain Tree Theorem is a classical result which expresses the stable distribution of an irreducible Markov matrix in terms of directed spanning trees of its associated graph. In this article, we present what we believe to be an original elementary proof of the theorem (Theorem 5.1). Our proof uses only linear algebra and graph theory, and in particular, it does not rely on probability...

It is well known that if the transition matrix of an irreducible Markov chain of moderate size has a subdominant eigenvalue which is close to 1, then the chain is ill conditioned in the sense that there are stationary probabilities which are sensitive to perturbations in the transition probabilities. However, the converse of this statement has heretofore been unresolved. The purpose of this art...

INTRODUCTION Andrei Andreyevich Markov was a Russian mathematician born in 1856. Markov's main interests were in number theory, continued fractions, and approximation theory. He studied under P. Chebyshev and worked in the field of probability theory. Markov pioneered work in the area of stochastic processes by creating a model called a Markov chain—a random walk on a state space. His work has ...

Let π denote an intractable probability distribution that we would like to explore. Suppose that we have a positive recurrent, irreducible Markov chain that satisfies a minorization condition and has π as its invariant measure. We provide a method of using simulations from the Markov chain to construct a statistical estimate of π from which it is straightforward to sample. We show that this est...

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which re...

Card shuffling models have provided simple motivating examples for the mathematical theory of mixing times Markov chains. As a complement, we introduce more intricate realistic model certain observable real-world scheme human players onto teams. We quantify numerically effectiveness this over 7 or 8 steps performed in practice. give combinatorial proof non-trivial fact that chain is indeed irre...

We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation σ of a Lebesgue probability space (X,A, μ) and show that these theorems are all equivalent. Two of these results concern the existence of a partition of the space X with special properties. The third theorem asserts that the conjugates of σ are dense in the uniform topology on the space o...

According to a 1975 result of T. Kaijser, if some nonvanishing product of hidden Markov model (HMM) stepping matrices is sub-rectangular, and the underlying chain is aperiodic, the corresponding α-chain has a unique invariant limiting measure λ. Here the α-chain {αn} = {(αni)} is given by where {(Xn, Yn)} is a finite state HMM with unobserved Markov chain component {Xn} and observed output comp...