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I am investigating a model, in which we are considering a discrete time gaussian random walk $(\Phi_k)_{k\geq{1}}$, which has state space $V\subset{\mathbb{R}^d}$. I have formed a tessellation of the subset $V$ by considering the Voronoi cells of a set of points in $V$ - we suppose we have the set of points $\lbrace{a,b,c,d}\rbrace$ with corresponding Voronoi cells $\lbrace{R_a,R_b,R_c,R_d}\rbrace$, where $R_a\cup{R_b}\cup{R_c}\cup{R_d} = V$ with the cells $R_i$ pairwise disjoint. I now want to construct a new process $(z_k)_{k\geq1}$ on the finite state space $\lbrace{a,b,c,d}\rbrace$,by letting $$z_k = a\mathbb{1}_{R_a}(\Phi_k)+b\mathbb{1}_{R_b}(\Phi_k)+c\mathbb{1}_{R_c}(\Phi_k)+d\mathbb{1}_{R_d}(\Phi_k)$$

In other words, at a time $k$ it describes whose Voronoi cell the Gaussian process is in. The main question I have is: Is the process Markov? And if so, how would one compute the transition matrix?

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    Not Markov, actually this procedure is the typical one used to destroy the Markov property (search for *hiden Markov models*). – 2017-02-22
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    Ok thanks, i'll have a look - if you have a link to an article which describes this procedure that would be great. – 2017-02-23
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    Did is correct, and probably also in this case (Gaussian process, tesselation). However, there are rare scenarios where your process is Markov. The keyword is *lumpability*, and there have been some papers by Murray Rosenblatt (Markovian Function of a Markov chain, etc.). – 2017-03-03

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