How many random walks correspond to a given set of return probabilities to the origin?
Markov chains on the space \(\{a, 0, 1, \ldots \}\) with positive transition probabilities \(0 \rightarrow 1 \rightarrow 2 \ldots \), \(0 \leftarrow 1 \leftarrow 2 \ldots\) and nonzero transition probabilities \(0 \rightarrow 0\), \(1 \rightarrow 1\), \(\ldots \), \(0 \rightarrow a\), \(1 \rightarrow a\), \(\ldots \), \(a \rightarrow a\) are examined. They are called in the title ``random walks''. If \(P^{n}_{00}\) is the return probability from the state \(0\) to \(0\) after \(n \geq 0\) steps, then there exists a unique probability measure \(\psi\) on \([-1, 1]\), called the spectral measure of the random walk, such that \(P^{n}_{00} = \int^{1}_{-1} x^{n} \psi (dx)\), \(n \geq 0\). Given a probability measure \(\mu\) on \([-1, 1]\) necessary and sufficient conditions for \(\mu\) to be the spectral measure of a random walk are given via a countable set of inequalities involving canonical moments of \(\mu\). If \(\mu\) satisfies the conditions, the class of all random walks having \(\mu\) for their spectral measure is characterized in a similar way. This class has one element if and only if \(\mu\) is the spectral measure of a recurrent random walk.
scientific article; zbMATH DE number 1095916