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Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

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Spectral representations of the transition probability matrices for continuous time finite Markov chains

Transition probability matrices for correlated random walks

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2 results

Spectral representations of the transition probability matrices for continuous time finite Markov chains

Transition probability matrices for correlated random walks