How many random walks correspond to a given set of return probabilities to the origin?
type: Item
instance of: scholarly article
title: How many random walks correspond to a given set of return probabilities to the origin?
zbMATH Open document ID: 0879.60067
DOI: 10.1016/S0304-4149(96)00083-X
published in: Stochastic Processes and their Applications
publication date: 1997-12-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 1095916
zbMATH Keywords: Markov chain
zbMATH Keywords: spectral measure
zbMATH Keywords: canonical moments
zbMATH Keywords: continued fractions
zbMATH Keywords: return probabilities
zbMATH Keywords: random walk
author: Holger Dette
author: William J. Studden
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1095916