The respective transition probabilities of two irreducible Markov chains 1 and 2 with common state space \(\mathbf{Z}=\{0,1, \ldots\}\) are for all \(i=0,1, \ldots\),

(1)

\(p_{i+1}=\frac{1}{i+2}, \quad p_{i 0}=\frac{i+1}{i+2} \quad\) and

(2) \(p_{i i+1}=\frac{i+1}{i+2}, \quad p_{i 0}=\frac{1}{i+2}\).

Check whether these Markov chains are transient, null recurrent, or positive recurrent.