Holroyd and Propp used Hall\u2019s marriage theorem to show that, given a probability distribution \u03c0 on a finite set S, there exists an infinite sequence s1, s2, . . . in S such that for all integers k \u2265 1 and all s in S, the number of i in [1, k] with si = s differs from k \u03c0(s) by at most 1. We prove a generalization of this result using a simple explicit algorithm. A special case of this algorithm yields an extension of Holroyd and Propp\u2019s result to the case of discrete probability distributions on infinite sets.<\/p>\n","protected":false},"excerpt":{"rendered":"
Holroyd and Propp used Hall\u2019s marriage theorem to show that, given a probability distribution \u03c0 on a finite set S, there exists an infinite sequence s1, s2, . . . in S such that for all integers k \u2265 1 and all s in S, the number of i in [1, k] with si = […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University Library","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2009-10-06","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/0910.1077","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13546],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-340532","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"Cornell University Library","msr_edition":"","msr_affiliation":"","msr_published_date":"2009-10-06","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"340535","msr_publicationurl":"https:\/\/arxiv.org\/abs\/0910.1077","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"stack","viewUrl":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-content\/uploads\/2016\/12\/stack.pdf","id":340535,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/0910.1077","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/0910.1077"}],"msr-author-ordering":[{"type":"text","value":"Omer Angel","user_id":0,"rest_url":false},{"type":"user_nicename","value":"holroyd","user_id":32020,"rest_url":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=holroyd"},{"type":"text","value":"James B. 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