Memoryless processes and independence

Question

this is a mere question of definition, that one surely can figure out by conventional means, but maybe someone can just quickly give me the definition.

What is a memoryless process?

Following the links on memoryless processes on Wikipedia, one arrives at the definition of memorylessness of a random variable. This however is a very different property from the memorylessness of a sequence of random variables, indexed by time. It is often said that Markov processes are memoryless. And vice versa? Do they mean that the Markov property is the memoryless property, i.e.

But then, I also often read in the context of B-processes that they are memoryless. Clearly B-processes are Markov, but with the additional property, that the actual random variables are all independent. So the probability of the present conditioned on any past is just the prob of the present. This would resemble the intuitive meaning for memorylessness the best, but I guess that's not what is meant, right?

Answer 1

Okay, so there are three definitions of memorylessness, each of which is incompatible with each other. That's crazy.

Def 1: A memoryless source is one in which each message is i.i.d Random variable. http://en.wikipedia.org/wiki/Information_theory

Def 2: A memoryless process $(X_n)$ (w.r.t prob space and filtration etc...) is one that satisfies the Markov property, i.e. $\mathbb P (X_n|X_{n-1},X_{n-2},X_{n-3},...)=\mathbb P (X_n|X_{n-1})$ http://en.wikipedia.org/wiki/Markov_process

Def 3: A memoryless random variable $X$ is one that satisfies \begin{equation} \mathbb P (X>m+n|X>m)= P(X>n) \end{equation} http://en.wikipedia.org/wiki/Memorylessness

So according to these definitions we have

1. A Markov process is a memoryless process
2. A memoryless source is a memoryless process
3. A memoryless process is NOT in general a memoryless source (i.e. a Markov processs has dependence between successive random variables)
4. A source of memoryless random variables is NOT in general memoryless.
5. A process of memoryless random variables is NOT in general momoryless.

o_O

Answer 2

I'm not entirely sure what it is you were asking after; What is a memory-less process? So, I shall stick to that portion.

If something is memory-less, what it means is future outcomes are not dependent on past outcomes. An example that is often used is waiting at a drive thru. If you've gone $30$ minutes without seeing a customer that doesn't mean you should expect one in the next $5$ minutes with any more likelihood than you would have $30$ minutes ago.

A more fun example, and one that is a good lesson to learn if you haven't already, is when you're at a roulette table. If you see it land on red $9$ times in a row, that doesn't increase the chance it will land on black on the next roll, or any roll for the matter. The probabilities are the same on every trial. Therefore, it is memory-less because past outcomes will not impact future ones.

Other examples: Coin Flips, Dice Rolls