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In this paper, we are the first to propose sound probabilistic extensions of LexRF with a weaker non-negativity condition, called <jats:italic>single-component<\/jats:italic> (SC) non-negativity. It is known that such an extension, if it exists, will be nontrivial due to the intricacies of the probabilistic circumstances.<\/jats:p><jats:p>Toward the goal, we first devise the notion of <jats:italic>fixability<\/jats:italic>, which offers a systematic approach for analyzing the soundness of possibly negative LexRSM. This notion yields a desired extension of LexRF that is sound for general stochastic processes. We next propose another extension, called <jats:italic>Lazy LexRSM<\/jats:italic>, toward the application to automated verification; it is sound over probabilistic programs with linear arithmetics, while its subclass is amenable to automated synthesis via linear programming. We finally propose a LexRSM synthesis algorithm for this subclass, and perform experiments.\n\n<\/jats:p>","DOI":"10.1007\/978-3-031-65633-0_19","type":"book-chapter","created":{"date-parts":[[2024,7,25]],"date-time":"2024-07-25T07:03:08Z","timestamp":1721890988000},"page":"420-442","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Lexicographic Ranking Supermartingales with\u00a0Lazy Lower Bounds"],"prefix":"10.1007","author":[{"given":"Toru","family":"Takisaka","sequence":"first","affiliation":[]},{"given":"Libo","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Changjiang","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Jiamou","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,7,26]]},"reference":[{"key":"19_CR1","unstructured":"Ultimate automizer. https:\/\/www.ultimate-pa.org\/?ui=tool &tool=automizer"},{"key":"19_CR2","doi-asserted-by":"crossref","unstructured":"Agrawal, S., Chatterjee, K., Novotn\u00fd, P.: Lexicographic ranking supermartingales: an efficient approach to termination of probabilistic programs. 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