'Markovian Property' vs 'Memoryless Property'

The two properties have the commonality in the sense that they predict the future based on the current state, not on the whole history of how the process wandered into the state. Then, what is the main difference btw two..? anyone can help me?

• I'd say Markov is more specific. For instance, sometimes we say that exponential random variables have the "memoryless property" that $P(X>t+s|X>s)=P(X>t)$. While this is related to continuous time Markov chains, it is not really a statement about stochastic processes.
– Ian
Commented Aug 23, 2015 at 14:29
• could you maybe add what you understand by these two properties? As Ian pointed out, the memoryless property could be understand in a non-markov way (clearly a markov process has not to fulfill $P(X>t+s|X>s)=P(X>t)$)... Commented Aug 23, 2015 at 15:08
• @Ian If the Memoryless property is a property of a random variable, and not a property of a stochastic process, then surely there's a way to generalize that and make it a property of a stochastic process, right? Commented Nov 22, 2018 at 6:52

I will try to explain the differences using the discrete case but it is the same idea in the continuous case.

Let $$(\Omega ,{\mathcal {F}},P)$$ be a probability space.

Let $$(X_n)_n$$ be a stochastic process (a sequence of random variables) taking value in a discrete space $$S = \{x_1,..,x_n\}$$. In that case, the markov property can be reformulated as follows:

Let $$n \in \mathbb{N}$$,

$$P(X_{n}=x_{i_n}\mid X_{n-1}=x_{i_{n-1}},\dots ,X_{0}=x_{i_{0}})=P(X_{n}=x_{i_{n}}\mid X_{n-1}=x_{i_{n-1}})$$

That means the probability the process takes a value $$x_{i_n} \in S$$ at time $$n$$ knowing all the previous values it has taken is the same as the probability the process takes a value $$x_{i_n} \in S$$ at time $$n$$ only knowing the previous value of the process.

Hence, if we are observing some random phenomenon (queue length evolution, stock-price..) and want to predict the next value of this process, the knowledge of the past since the very beginning ($$X_{n-2}=x_{i_{n-2}},\dots ,X_{0}=x_{i_{0}})$$ is not necessary, only $$X_{n-1}=x_{i_{n-1}}$$ matter to evalutate the probabilities the random variable $$X_n$$ could take, provided what we are observing has the markov property.

Further more, let explain what is the memorylessness property. Let $$X$$ be a random variable taking value in $$\mathbb{R}_+$$. We say $$X$$ has the memoryless property if:

$$P(X>t+s\mid X>t) = P(X>s)$$ for any non-negative real numbers $$t$$ and $$s$$.

First, one can note that there is one only random variable involved in this definition. What should be understood in that definition is that the probability of $$X$$ being above some treshold $$s$$ is the same as the probability of $$X$$ being at a distance $$s$$ of $$t$$ knowing $$X$$ is already above $$t$$. Knowing the state of the phenomenon $$X$$ (being superior than $$t$$) is not helpful to evaluate something which depends on that given state. Remembering the state of X (the knowledge that the event $$\{ X > t \}$$ has already occurred) in this case is useless.

For instance, knowing you have waited at the bus station for 0 min the probability that the bus will come after 7 min could be the same as the probability the bus come after 22min if you came at the minute 15 or the probability the bus come after 37min if you came at the minute 30. In this scenario only the distance from the bus arrival time matter.

Finally, the markov property is path-related whereas the memoryless property is just describing the behavior of one fixed phenomenon represented by $$X$$.

Markov process are memoryless in the sense that you only need to know the current state in order to determine statistics about its future. The past does not impact statistics about the future.

Now, you can encompass as much information as you wish in the state of a Markov process. Therefore, in a Markov chain, for instance, you can store information about the state of the system in the past few days in order to determine, for instance, if it will rain or not tomorrow.

Positive perspective towards Markov processes: Markov chains are much more powerful than exponential random variables. Both subsume Markov processes, where the latter subsumes the simplest possible Markov process. Markov chains can encapsulate the whole history in its state. Exponential random variables have no state at all.

Negative perspective towards Markov processes: if you want to store information about the past $$n$$ days, you need a Markov chain with $$2^n$$ states. The cardinality of the state space grows exponentially with respect to $$n$$.

Finally, note that n-grams, for instance, illustrate a canonical example of the distinction above between Markov processes and the simplest possible memoryless processes.

When people say that the sole memoryless distribution is the exponential distribution (resp., geometric for discrete case) they are referring to the fact that all other state-less distributions do not have the memoryless property. They are not accounting for the fact that Markov chains are also memoryless, given that Markov chains are state-full.