How to characterize recurrent and transient states of Markov chain According to Wikipedia with a little rephrasing:

A state $i$ is transient if and only
  if $P(T_i < \infty) <1$, recurrent if
  and only if $P(T_i < \infty) =1$,
  where $T_i$ is the first hitting time
  to i, i.e. $T_i=\inf\{n \in \mathbb{N} \cup \{ \infty \}: X_n=i \mid X_0=i \}.$

If I understand correctly, this can be used as the definition of transient/recurrent state. 
Usually $P(T_i < \infty)$ is written as a series $\sum_{n \in \mathbb{N}} P(T_i = n)$. But I would like to learn other ways to tell if a state is recurrent/transient, which might be easier in some cases.


*

*For example, can a
transient/recurrent state be
completely characterized in terms of
closed subsets of states (defined similarly as an absorbing state), as follows
(my own quote)?

State $i$ is transient if and only if
  there exists a closed subset $S$ of
  states, s.t. $i \notin S$ and there exists $s \in S$
  and $n \in \mathbb{N}$ and the
  $n$-step transition probability
  $p_{is}^{(n)} > 0$.
Similarly, State $i$ is recurrent if
  and only if there does not exist such a
  closed subset of states as described above?

Can we also characterize positive/null
recurrence in terms of closed
subsets of states?

*Off the top of your head, what are some other necessary and/or sufficient
conditions for recurrent/transient
and positive/null recurrent state?


Thanks and regards!
 A: (0) The definition of $T_i$ on Wikipedia is awful on at least three counts. One should define $T_i=\inf A_i$ with $A_i=\{n\ge1:X_n=i\}\cup\{+\infty\}$ and one should say that $i$ is transient if and only if $P(T_i<+\infty|X_0=i)<1$ and recurrent otherwise.
(1) In some cases there exists no closed subset at all and the existence of closed sets is not related to recurrence or transience. 
First example: consider a homogenous random walk on $\mathbb{Z}$. Thus $p_{n,n+1}=a$ and $p_{n,n-1}=1-a$ for every $n$, for a given $a$ in $(0,1)$. Then there exists no closed set except $\mathbb{Z}$ and the chain is recurrent if $a=\frac12$ and transient otherwise. 
Second example: consider a homogenous birth-and-death chain such that $0$ is absorbing. Thus $p_{0,0}=1$ and $p_{n,n+1}=a$ and $p_{n,n-1}=1-a$ for every $n\ge1$, for a given $a$ in $(0,1)$. Then the set $S=\{0\}$ is closed and the chain is recurrent if $a\le\frac12$ and transient otherwise.
(2) You could (should?) read the beautiful small book Random Walks and Electric Networks by Peter G. Doyle and J. Laurie Snell, which explains this and a lot of related stuff in a very accessible way.
A: *

*Tim's characterization of states in terms of closed sets is correct 
for finite state space Markov chains. Partition the state
space into communicating classes. Every recurrent class is closed, 
but no transient class is closed (because the chain must eventually
get "stuck" in some recurrent class). The part in parentheses is false
for infinite state space chains, as Didier's answer shows. 

*Another well-known characterization is that a state $i$ is transient
if and only if $$\sum_{n=1}^\infty P(X_n=i | X_0=i)<\infty.$$ This criterion is used,
for example, to prove Polya's result that the symmetric random walk on  $\mathbb{Z}^d$ is recurrent if $d=1,2$, but transient when $d\geq 3$.   

*Similarly, the probability
$$P(X_n=i\mbox{ for infinitely many } n | X_0=i)$$ 
is equal to zero or one, depending on whether the state $i$ is transient or recurrent.  
