I am facing a problem regarding the stationary distribution of mobility models in revising my paper.
In a general sense, my question is like this.
- Given a stochastic process {Xt: t>=0} such that the initial state X0 is uniformly distributed in the state space and for all t > 0, Xt = X0,namely, the process does not evolve with time. For this specific process, can we say that {Xt} has a stationary distribution that is uniformly distributed in the state space?
Hi,
of course I do not know the specific framework of your question, but a stochastic process for which X_t = X_0 for all t is simply not a process ! A stationary distribution is defined as the unique limit distribution obtained for t -> infinity. The problem I see with your 'process' is, that any initial distribution will be the limit distribution for t -> infinitiy, thus it is not unique and therefore the term 'stationary distribution' is not correct.
The answer is quite simple: Yes.
The process "inherits" the initial distribution for all time instants through the perfect correlation, so the initial distribution equals the stationary distribution. And your process is perfectly well defined.
In a markovian process the asymptotic distribution does not depend on the initial distribution. It is the essence of the Markov process: it does not keep memory on its past. More of that, if you deal with a discrete sample space (with a Markov chain) and its evolution is described by the matrix of transition probabilities, the asymptotic distribution is the eigen-vector of the matrix corresponing to its eigen-value equal to 1. The initial distribution is not involved into this operation.
Markovian processes have a unique limiting distribution only if they are irreducible, otherwise they in general have one limiting distribution for each of the infinite possible initial distributions.
In this case the limiting distribution corresponds to the initial distribution, so if the initial distribution is uniform the limiting distribution will be as well.
For a Markov chain in discrete time and stationary transition probability matrix and a set of communicating states the process will converge in distribution to a stationary distribution whatever the initial state random or otherwise.
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