In many books Ehrenfest Urn is used as an example of a homogeneous Markov chain, where entries in transition probabilities depend on the state, e.g.
$$ p_{i,i+1} = \frac{i}{n}, q_{i,i-1} = 1- \frac{i}{n} $$
or, in recurrent notation, for period $t$:
$$ S_{t} = S_{t-1} + \xi_{t-1} $$ where $\xi_{t-1}$ is a random variable. What I have never seen is the proof that this process satisfies weak Markov property:
$$ P(S_{t+1} = i_{t+1}|S_{t} = i_{t},...,S_{0}=i_{0}) = P(S_{t+1} = i_{t+1}|S_{t} = i_{t}) $$
Only in Flajolet, Dumas,Puyhaubert(2006) on p.70 it is mentioned that Ehrenfest urn can be viewed as a random walk on N-dimensional cube, but I can't relate it.
I'd massively appreciate suggestion on how to approach this proof.
