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### Prove that $P(X_{n+m}=j|X_0=i_0,\dots, X_n=i)=p_{ij}^{(m)}$

I will sketch the proof of the case $m=2$; the general case is analogous. Let $E_{n,i} = \{X_0=i_0, \ldots, X_n=i\}$. \begin{align} P(X_{n+2} = j \mid E_n) &= \sum_{k} P(X_{n+2} = j, X_{n+1} = k \...
• 90k
Accepted

### A question about Markov chain's definition

It is not necessary: if you know that $$P(Z_{n+1}=y \mid Z_0=x_0,\dots,Z_{n-1}=x_{n-1},Z_n=x)=P(Z_1=y \mid Z_0=x)$$ then you can show that both of those are equal to $P(Z_{n+1}=y\mid Z_n = x)$. To ...
• 143k
1 vote

### Demonstrate a (continuous time) chain that increases by one, and then randomly returns to the origin, is transient

Say we start at state $0$. What events need to happen for us to never return to $0$? For this particular chain, there's only one path that doesn't return. From $0$, we need to step to $1$ (prob. $1$)....
• 86
1 vote
Accepted

### Defining majorization of vectors using doubly stochastic matrix

The following proof is given in Bhatia's Matrix Analysis as part of the proof for Theorem II.1.9. Let $A$ be doubly stochastic and let $y = Ax$. To prove that $y \prec x$, we may assume without loss ...
• 226k
1 vote

### Defining majorization of vectors using doubly stochastic matrix

If $Da=b$ with $D=(p_{ij})_{1\leq i,j\leq d}$ bistochastic then for fixed $1\leq k\leq d-1$, $c_j=\sum _{i=1}^kp_{ij}\leq 1$ for $j\leq d.$ Since $D$ is bistochastic we have $\sum_{j=1}^dc_d=k$. ...
• 1,125
1 vote

### Proving The Fundamental Theorem of Markov Chains

One reference which has this proof is Introduction to Probability for Computing by Harchol-Balter, specifically Theorem 25.12 [PDF link].
• 86

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