All Questions
8 questions
9
votes
2
answers
953
views
Direct proof of unique invariant distribution for ergodic, positive-recurrent Markov chain
I posted the following question on MSE, feeling that it perhaps isn't research level mathematics, but didn't get any bites. So, I am crossposting here.
The following ergodic theorem is well known.
...
- 165
3
votes
0
answers
182
views
Spectral radius of infinite substochastic upper triangular matrix
Let $M$ be a Markov chain on $\{0, 1, 2, \dots\} \cup \{\delta\}$, where $\Pr(i \to j) > 0$ for $i, j \in \mathbb{N}$ only if $j > i$, and $\Pr(\delta \to \delta) = 1$. This represents a birth-...
- 131
3
votes
0
answers
158
views
Worst-Case Solution to (Stochastic) Matrix Inequality
EDIT: Some specific conjectures added.
This problem comes with an associated stochastic process, but I phrase everything as linear algebra in case somebody from a non-probability community has seen ...
- 31
3
votes
0
answers
770
views
Kullback-Leibler Divergence of Stationary Distributions of Markov chains
Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...
- 418
5
votes
1
answer
476
views
Elementary Markov Chain Question
Are any general conditions known on a finite transition nxn matrix that ensure that there exists at least one mth root which is also a transition matrix? It is easy to construct a 3x3 , diagonally ...
- 313
1
vote
2
answers
307
views
Continuous family of Markov chains
Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition ...
- 1,069
2
votes
2
answers
861
views
Spectral gap of a product of Markov processes
For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [...
- 15.4k
10
votes
1
answer
1k
views
Bounds on $\|P^{k+1} - P^k\|$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k\gg n$
The problem:
We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
- 303