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Consider the transition matrix

$ P = \begin{bmatrix} 1-p&p\\ q&1-q \end{bmatrix} $

for general $2$-state Markov Chain $(0 \le p, q\le 1)$.

(a) Find the limiting distribution (if it exists) if $p + q = 1$.
(b) Find the limiting distribution (if it exists) if $p + q \ne 1$.

I myself calculated (a) with the answer $(\frac{q}{p+q}, \frac{p}{p+q}) = (q, p )$.

enter image description here

But, I couldn't understand the problem (b).

If $p+q \ne 1$, then the answer should be $(\frac{q}{p+q}, \frac{p}{p+q})$. Isn't that so?

What else could have been derived?

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  • $\begingroup$ Well what if $p=q=1$ does the matrix converges $P^n$? $\endgroup$ – Phicar Apr 12 at 18:02
  • $\begingroup$ In your solution, did you actually use the fact that $p + q \neq 1$, other than in the very last step, when you resolve $(\frac{q}{p+q}, \frac{p}{p+q})$ to be $(p, q)$? I think you're right that the two solutions are nearly identical and can both be seen as specific cases of not placing any constraints on $p, q$ (besides $0 \leq p, q \leq 1)$. $\endgroup$ – Aaron Montgomery Apr 12 at 18:21
  • $\begingroup$ @AaronMontgomery, kindly, see the edit. $\endgroup$ – user366312 Apr 12 at 19:37
  • $\begingroup$ @Phicar, kindly, see the edit. $\endgroup$ – user366312 Apr 12 at 19:37
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Since:

$$\begin{bmatrix} q&p \end{bmatrix} \begin{bmatrix} 1-p&p\\ q&1-q \end{bmatrix} =\begin{bmatrix} q - pq + pq & qp + p - pq \end{bmatrix} = \begin{bmatrix} q&p \end{bmatrix} $$

We have $v = [q\ \ p]$ is a right eigenvector with eigenvalue $1$, and the limiting distribution is a rescaled version of $v$ whose entries sum to $1$, i.e. $\pi = {v \over \sum_i v_i} = [{q \over p+q}\ \ {p \over p+q}]$.

None of the above depends on the value of the sum $p+q$; i.e. it is valid whether $p+q = 1$ or $= \sqrt{2}$ or $= 10^{-4}$ or any other value.

Nice trick question though!


Ooh, as @Phicar pointed out, the case of $p=q=1 \implies P^n$ does not converge. If you define the "limiting distribution" as $\lim P^n \pi_0$ then it does not exist. But if you define the "limiting distribution" as the unique probabilistic vector s.t. $\pi P = \pi$ then it does exist (and $\pi = [{q \over p+q}\ \ {p \over p+q}]$ as always). Not sure the terminology in this corner case...

Come to think of it, the $p=q=0$ case is even more problematic as any $\pi P = \pi$ (since $P=$ identity matrix).

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  • $\begingroup$ have u seen the solution in the question? Can u kindly explain it? $\endgroup$ – user366312 Apr 12 at 19:42
  • $\begingroup$ the "solution" meaning the scanned handwritten note? i didn't check the details but that looks like a direct exact derivation of $P^n$ via eigen decomposition, maybe? anyway, if $0 < p+q < 2$ then $|1 - p - q| < 1$ and so $\lim (1 - p - q)^n = 0$ so all those terms vanish, and you can clearly see $P^\infty$ having two identical rows, both equal to $\pi$. $\endgroup$ – antkam Apr 12 at 19:49
  • $\begingroup$ @antkam Being limiting distribution implies being stationary. Not backwards. $\endgroup$ – Phicar Apr 12 at 19:51
  • $\begingroup$ @Phicar - thanks, I was not sure of the terminology (since I learned this literally decades ago). Then in the $p=q=1$ case there is no limiting distribution, while in the $p=q=0$ case every distribution is limiting. Right? $\endgroup$ – antkam Apr 12 at 19:55
  • $\begingroup$ Are we mixing up Limiting Distribution and Stationary Distribution? $\endgroup$ – user366312 Apr 12 at 20:03

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