# States of Markov chain and stationary distribution

Let $$X$$ be a Markov chain with a state space $$S={\{0,1,2,... \}}$$ and a transition matrix $$P$$ with given $$p_{i,0}=\frac{i}{i+1}$$ and $$p_{i,i+1}=\frac{1}{i+1}$$, for $$i=0,1,2,...$$. Find out which states are transient, null and not-null. Find stationary distribution.

I have that $$p_{0,0}=0, p_{1,0}=\frac{1}{2}, p_{2,0}=\frac{2}{3}, p_{3,0}=\frac{3}{4}, ...$$ and $$p_{0,1}=1, p_{1,2}=\frac{1}{2}, p_{2,3}=\frac{1}{3}, p_{3,4}=\frac{1}{4},...$$.

All the states are persistent so no states are transient

A state is null $$\iff \lim_{n \to \infty} p_{i,i}(n)=0$$, so state $$0$$ is null (because $$\lim_{n \to \infty} \frac{1}{n+1}=0$$).

From now I don't know how to determine whether states $$1,...,J$$ are null or not-null. My idea was to let $$T_i$$ be the time of first return to $$i$$, but I'm not sure how to continue.

• Please add the self-study tag since this is related with an exercise. – Xi'an Jan 12 at 14:25