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The corresponding Hessenberg matrix can be normalized to a stochastic matrix. We find twelve uniform Hessenberg matrices of Toeplitz type and their normalizations to stochastic or semi-stochastic matrices. The corresponding weights and the stochastic bidiagonal factorizations for these matrices are provided. Only one of these uniform stochastic cases, which is a recurrent Markov chain, has a uniform stochastic factorization. Chain of Christoffel transformations connecting the stochastic uniform tuples and the semi-stochastic uniform tuples, between them, are presented.<\/jats:p>","DOI":"10.1007\/s11075-025-02195-6","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T12:44:50Z","timestamp":1754397890000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Uniform multiple orthogonal polynomials and Markov chains"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4685-1583","authenticated-orcid":false,"given":"Am\u00edlcar","family":"Branquinho","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7932-3712","authenticated-orcid":false,"given":"Juan E. 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