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• 26.7k
Accepted

For which values of $a$ is this Markov Chain irreducible and aperiodic?

For the first bit, yes, you are essentially right. To prove this for small chains, I like to enumerate the states (say, as $1,2,3,4,5$), then (assuming you start in the $i$th row and move to the $j$th ...
• 45.1k

Decomposing a general stopping time into stopping components

(The $y$ in Frid's notation is misleading.) Instead, for a stopping time $\tau\ge 1$ and a state $x$, define $T_x(\omega):=\tau(x\omega)-1$. (Here, for a path $\omega=a_0a_1a_2\cdots$, the ...
• 26.5k
1 vote
Accepted

Are Markov chains $X_{t+r}$, $X_{2t}$ and $(X_t; X_{t+1})$?

Since $X_k$ is a Markov chain we have $$\mathbb{P}(X_{2n+1}=k\mid X_{2n}=i_0, X_{2n-2}=i_2,\dots) =\mathbb{P}(X_{2n+1}=k\mid X_{2n}=i_0)$$ Next,  \mathbb{P}(X_{2n+2}=i\mid X_{2n}=i_0, X_{2n-2}=i_2,...
• 3,367
1 vote
Accepted

Coupling identical Continuous-Time Markov Chains

First, I assume the state space, say $E$, is countable since you are using indices $i,j$ to denote arbitrary states. This assumption is crucial for our discussion. 1. Positive Recurrence An ...
1 vote
Accepted

Finding References for known results about mixing time of a Markov chain arising from Gaussian elimination

There are at least $N=c2^{n^2}$ invertible matrices mod 2, where $c=1/2(1-1/4)(1-1/8)\cdots (1-2^{-k})\cdots>0$. Since there are less than $n^2$ possible moves in each step, after $t$ steps the ...
• 22.2k

Proof of Chapman-Kolmogorov equations for the general case

Let $x_t$ be the state $t \geq 0$ period ahead \begin{align*} Q^{n+m}(x,\,A)& = \int Q^{n+m-1}(x_1,\,A)\,Q(x,\,d\!\,x_1)\\ & = \int \Bigl(\int Q^{n+m-2}(x_{2},\,A)\,Q(x_1,\,...
Accepted

circular random walk - markovian frog

The set of states the frog has visited is always a contiguous block. It can grow on both ends as the frog can traverse back and forth in the visited block. That block needs to grow $n-1$ times. The ...
• 9,548