# Confusion Regarding the Definition of a Markov Chain

Wikpedia defines a discrete-time Markov chain as a sequnce of random variables $X_1, X_2, X_3, \ldots$ with the property that $$P(X_{n+1}=x| X_1=x_1, X_2=x_2, \ldots, X_n=x_n) = P(X_{n+1}=x_{n+1}| X_n=x_n)$$

if both conditional probabilities are well-defined. Here all the random variables $X_i$ take values in a countable set $S$.

I do not understand what is the meaning of the above definition. A random variable has its domain as a probability space. The definition makes no mention of any probability space. I checked K. L. Chung's Elementary Probability Theory, and there also the definition is stated in the same manner, wihtout any reference to a probability space.

• The probability space is there, just not mentioned. What do you think $P(X_n = x_n)$ means, for example? – mathworker21 Feb 25 '18 at 6:50
For instance, how many problems have you done with a premise like "let $X$ and $Y$ be independent standard normal random variables"? There's a probability space lurking behind here, no doubt, but it would be a waste of time to define it. We know that it's possible to build probability spaces on which a pair of random normals can be defined (for instance the probability space defined by the joint law of the random variables works... but we're by no means committed to it if we want to later consider a third RV $Z$).