A discrete-time Markov chain is a sequence of random variables $$X_1, X_2, X_3, \ldots$$ with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states
$$Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),$$ if both conditional probabilities are well defined, i.e. if $$Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.$$