TITLE: Relation between Memorylessness and Independence
QUESTION [6 upvotes]: I was reading about Memorylessness and Independence and was wondering whether there is any relationship between the two. I realize that memorylessness is a property of a random variable of a particular distribution (such as X is a RV following exponential distribution) and independence has more to do with 2 or more random variables rather than their distribution.
But, is there any fact that links the two?

REPLY [2 votes]: The lack of memory or Markov property is a form of conditional independence of RVs.
Consider X$_i$  i=1,2,...t where increasing index may be viewed as moving forward in time.
Memoryless property
P[X$_t$=x| X$_t$$_-$$_1$=y$_1$, X$_t$$_-$$_2$=y$_2$, ...X$_1$=y$_t$$_-$$_1$] =
P[X$_t$=x| X$_t$$_-$$_1$$=y$$_1$] for any x and y$_1$,...y$_t$$_-$$_1$
In words this means X$_t$ is INDEPENDENT of 
X$_t$$_-$$_2$ , ..., X$_1$ CONDITIONAL ON X$_t$$_-$$_1$