# Markov property definition

The definition of the Markov property is typically that the next state depends only on the present state and no past states. However, the mathematical definition I usually see (e.g. https://stats.stackexchange.com/a/2463/225553) says that P(X_t = x | X_{t-1}, X_{t-2}, X_{t-3}...X_t0) = P(X_t | X_{t-1}). That seems to say that the present state depends on only the immediate prior state. Are these equivalent definitions or am I misunderstanding the notation?

You correctly wrote... $$P(X_t = x | X_{t-1}, X_{t-2}, X_{t-3},...X_{t0}) = P(X_t | X_{t-1})$$ However this is equivalent to the following by virtue of shifting the time index by 1. $$P(X_{t+1} = x | X_{t}, X_{t-1}, X_{t-2},...X_{t0}) = P(X_{t+1} | X_{t})$$
• Thanks for answering. The shifted time index makes sense. I'm still a bit confused. Does this mean that $$P(X_t = x | X_{t-1}, X_{t-2}, X_{t-3},...X_{t0}) = P(X_t | X_{t-1})$$ $$X_{t-1}$$ is considered the "present"? I am still unsure if the statements "The future state depends only on the present state" is equivalent to "The present state depends only on the immediate past state". Nov 2, 2018 at 5:29
• @coderunner, if you were at $t-1$ in time then at that moment it would be the present, right? Similarly, if you were at $t-5$ in time, $t-6$ would be the immediate past for the process and $t-4$ would be the future. I'm not sure if that helps... Nov 2, 2018 at 5:33