{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:48:09Z","timestamp":1740491289859,"version":"3.38.0"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"A","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2014,12]]},"abstract":"Let \ud835\udcc8<\/jats:italic> be a finite or countable set. Given a matrix F<\/jats:italic> = (F<\/jats:italic>\n \n ij<\/jats:italic>\n <\/jats:sub>)\n i<\/jats:italic>,j<\/jats:italic>\u2208\ud835\udcc8<\/jats:italic>\n <\/jats:sub> of distribution functions on R<\/jats:bold> and a quasistochastic matrix Q<\/jats:italic> = (q<\/jats:italic>\n \n ij<\/jats:italic>\n <\/jats:sub>)\n i<\/jats:italic>,j<\/jats:italic>\u2208\ud835\udcc8<\/jats:italic>\n <\/jats:sub>, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u<\/jats:italic> and v<\/jats:italic>, the matrix renewal measure \u2211\n n<\/jats:italic>\u22650<\/jats:sub>\n Q<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sup> \u2297 F<\/jats:italic>\n *n<\/jats:italic>\n <\/jats:sup> associated with Q<\/jats:italic> \u2297 F<\/jats:italic> := (q<\/jats:italic>\n \n ij<\/jats:italic>\n <\/jats:sub>\n F<\/jats:italic>\n \n ij<\/jats:italic>\n <\/jats:sub>)\n i<\/jats:italic>,j<\/jats:italic>\u2208\ud835\udcc8<\/jats:italic>\n <\/jats:sub> (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q<\/jats:italic> \u2297 F<\/jats:italic> becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q<\/jats:italic> \u2297 F<\/jats:italic> to a Markov random walk {(M<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>, S<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>)}\n n<\/jats:italic>\u22650<\/jats:sub> with discrete recurrent driving chain {M<\/jats:italic>\n \n n<\/jats:italic>\n <\/jats:sub>}\n n<\/jats:italic>\u22650<\/jats:sub>. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.<\/jats:p>","DOI":"10.1017\/s0021900200021380","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T16:25:31Z","timestamp":1459355131000},"page":"359-376","source":"Crossref","is-referenced-by-count":0,"title":["Quasistochastic matrices and Markov renewal theory"],"prefix":"10.1017","volume":"51","author":[{"given":"Gerold","family":"Alsmeyer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,3,30]]},"reference":[{"key":"S0021900200021380_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/0025-5564(71)90058-7"},{"key":"S0021900200021380_ref27","first-page":"595","volume":"74","year":"2010","journal-title":"Izv. Ross. Akad. Nauk Ser. Mat. 74, 157\u2013168 (in Russian). English translation: Izv. 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