An irreducible, aperiodic and recurrent Markov chain

Question

Suppose $X_n,n=0,1,2,...$ is a Markov chain with transition probability $(p_{ij})$. Let $p_{0j}>0,j=0,1,2,...$ and for all $i \geq 1$ $$\sum^{\infty}_{j=0}jp_{ij}\leq i-\frac{1}{10}.$$ I have no idea to prove that $X_n$ is irreducible, aperiodic and recurrent. Can someone give me a hint?

Answer 1

Hint: The left hand side is an expectation. Use the inequality show that for every $i \geq 1$, we have $p_{ij} \neq 0$ for some $j < i$. From there, it's easy to show that the markov chain is irreducible and recurrent.

We can also show that the period at state zero, namely $\gcd\{n>0 : \mathbb P(X_n = 0\mid X_0 = 0) > 0\}$, is $1$. Because we know that the chain is irreducible, this is enough to deduce that the chain is aperiodic.