I was reading the well-known paper by Warren Weaver, "Recent Contributions to The Mathematical Theory of Communication", I stumpled upon the following sentence(p. 5)"
A system which produces a sequence of symbols (which may, of course, be letters or musical notes, say, rather than words) according to certain probabilities is called a stochastic process , and the special case of a stochastic process in which the probabilities depend on the previous events, is called a Markoff process or a Markoff chain.
I am aware that Markoff chains are indepdent on the previous states of the system, formally :
$Prob(X_{k} = j_{k} / X_{k-1} = j_{k-1} ... X_{0} = j_0) = Prob(X_{k} = j_{k} / X_{k-1} = j_{k-1}) ~(*)$
Where, obviously, the chain consists of a finite set of states $S$ and the transition probabilities from state i to j $P_{ij}$ are characterized by (*)
Am I missing something?
The author is trying to describe one of the distinctive qualities of Markov chains.
In many stochastic processes, the value of the next symbol is entirely independent of the previous one. If, for example, we have $x_n$ where $x_n \in \{1, 2, 3, 4, 5, 6\}$ (eg outcomes when a die is thrown), then the value of $x_{n+1}$ is independent of $x_n$. In a Markov process, however, the possible values of $x_{n+1}$ do depend on the current value of $x_n$ and the probabilities will be different, too. So we could have a situation where $x_n=1$ can be followed only by $x_{n+1}=3$ (with probability 30%) or $x_{n+1}=6$ (with probability 70%) and $x_n=2$ can be followed by $x_{n+1}=2$ (with probability 50%) or $x_{n+1}=3$ (with probability 50%). Thus both the probabilities and the possible next events depend on the previous events.