0
votes
1answer
72 views

Convergence rate of PageRank, the problem when the second eigenvalue is complex

As far as I know the Google matrix used to calculate the PageRank is not symetric, that means that some eigenvalues can be complex, furthermore, we know that the second eigenvalue is equal to the ...
0
votes
1answer
33 views

Why does markov chains power method converge at the rate of |λ_2/λ_1 |

I'm doing some researches on Markov Chains, and every time I meet this statement, that The rate of convergence of the power method is given by |λ_2/λ_1 |^k→0, when k→inf. And where λ_1 and λ_2 are ...
0
votes
0answers
40 views

Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
0
votes
0answers
30 views

Convergence of sequence of stationary distributions of Markov chains

I have a sequence of finite, discrete-time ergodic Markov chains indexed by a parameter $N$, and I want to prove that their stationary distributions are converging to a well-defined limit as $N\to ...
1
vote
4answers
173 views

Why do these expressions tend to zero?

In a chapter on Markov chains, it is claimed that $\binom {2n}{n} p^n(1-p)^n$ (where $p$ is a probability) tends to $0$ as $n$ tends to $\infty$. But why is this so? It is also claimed that for an ...
3
votes
0answers
114 views

Monotonic convergence of powers of a stochastic matrix

Let $P$ be a stochastic matrix (nonnegative and each row summing to 1). Assuming that $P^n$ converges to $\textbf{1}\pi$ as $n \rightarrow \infty$, where $\pi$ is a row vector (stationary distribution ...
3
votes
0answers
95 views

maximum renewal rate of a Markov chain

Consider a Markov chain $(X_t)$ on a state-space with a countably generated $\sigma$-algebra and assume this Markov chain allows for small sets of order one. This means there exist sets $\mathfrak{S}$ ...
2
votes
2answers
61 views

Convergence of an Ergodic process

I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
1
vote
1answer
471 views

Expected first hitting time in an absorbing Markov chain

There exists an absorbing $(M \times M)$Markov chain with the following transition matrix: $$ \begin{array}{ccccccc} p_{11} & p_{12}&0&\cdots&\cdots&\cdots&0\\ p_{21} & ...
2
votes
2answers
285 views

What does it mean for MCMC to converge?

I know that a Markov Chain is a discrete random process where the current state decides the next and in a random walk, the probability that we move from node u to v is 1/N(u). An MCMC sample will ...