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A lumping is strongly k<\/jats:italic>-lumpable, if and only if the lumped process is a k<\/jats:italic>th-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k<\/jats:italic>-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k<\/jats:italic>-lumpable.<\/jats:p>","DOI":"10.1017\/s0021900200012018","type":"journal-article","created":{"date-parts":[[2016,3,29]],"date-time":"2016-03-29T14:51:11Z","timestamp":1459263071000},"page":"1114-1132","source":"Crossref","is-referenced-by-count":5,"title":["Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability"],"prefix":"10.1017","volume":"51","author":[{"given":"Bernhard C.","family":"Geiger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christoph","family":"Temmel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,4]]},"reference":[{"key":"S0021900200012018_ref19","first-page":"92","volume":"5","year":"1978","journal-title":"Scand. J. 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