Let $m_{1,0}$ denotes the expected time of transitions starting from state $1$, until a transition into state $0$ occurs, in one dimensional symmetric random walk, with infinite countable states i.e. state $0,\pm 1, \pm2,...$
In the one dimensional symmetric random walk, it has been proven that: the mean number of transitions to go from state $1$ to either $0$ or $n$ is $n − 1$
From state $1$ to state $n$, there are at least $n-1$ transitions and $n-1 = m_{1,0}$*Pr{getting to $0$}+$m_{1,n}$*Pr{getting to $n$}
so I think is should be $m_{1,0}\le n-1$ because the mean number is $n-1$.
But in the book: Introduction to Probability Models (Sheldon M. Ross) $~~$page 218, there is:
The classical example of a null recurrent Markov chain is the one dimensional symmetric random walk of Example 4.18. One way to show it is null recurrent is to argue that the mean time to return to a state is infinite.
(For another argument, see Exercise 39.) To show this, let $m_{i,j}$ denote the mean number of transitions, starting in state i, until a transition into state j occurs.
Now, in Example 3.16, it was shown that the mean number of transitions to go from state 1 to either $0$ or $n$ is $n-1$, implying that $m_{1,0}\ge n-1$
So I wonder which one is wrong.
The book is correct. Getting to $0$ takes at least as long as getting to either $0$ or $n$, so the expected time to get from $1$ to $0$ is at least the expected time to get from $1$ to either $0$ or $n$.