Limiting distribution of an infinite Markov Chain

Question

Let the following infinite matrix P represent an Infinite States Markov Chain.

\begin{pmatrix} 1 & 0 & 0 & 0 & \cdots\\ 1 & 0 & 0 & 0 &\cdots\\ 0 & 1 & 0 & 0 & \cdots\\ 0 & 0 & 1 & 0 & \cdots\\ 0 & 0 & 0 & 1 & \cdots\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}

My purpose is to find its limiting distribution.

So, as far as I know, I have to find a vector x= (x(1),x(2),x(3),x(4),...) wich satisfies: xP=x and x(1)+x(2)+x(3)+...+x(i)+...= 1

That x will be a stationary distribution for this chain.

That gives me the following equations:

x(1) + x(2) = x(1)

      x(3) = x(2)
           .
           .
           .
    x(n+1) = x(n)

Which results in:

x(1)=1

x(2)=x(3)=...=x(n)=...=0

So the stationary distribution is unique and it´s : x=(1,0,0,0,0,0,0,0,0,0,0...)

How can I know it is also the limiting distribution?

(sorry for the rude writting, I am new and some text commands did´t work)

Answer 1

Take any finitely supported initial distribution $p$; say the maximum of the support is at $N$. Then $p P^{N-1}$ is already the stationary distribution $\pi$. Now given any arbitrary initial distribution $q$ and $\epsilon>0$, $q$ is within $\epsilon$ in the $\ell^1$ metric of some finitely supported distribution $q_\epsilon$.

Lemma: for any infinite row-stochastic matrix $P$ and any row vectors $p,q \in \ell^1,\| pP - qP \|_{\ell^1} \leq \| p - q \|_{\ell^1}$.

Therefore there is $N(\epsilon)$ such that $\pi=q P^{N(\epsilon)-1}$, hence $q_\epsilon P^{N(\epsilon)-1}$ is within $\epsilon$ of $\pi$.