{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19493"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19493","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Bounds on the size of strong subordinates of submartingales and subharmonic functions","abstract":"Suppose X is a submartingale that is continuous on the right with limits from the left and H is a predictable process bounded by 1 in absolute value. Let $Y = (Y\\sb{t})\\sb{t\\ge 0}$ where$$Y\\sb{t} = H\\sb0X\\sb0 + \\int\\sb{(0,t\\rbrack} H\\sb{s}dX\\sb{s}.$$An interesting and important question is: How large is Y compared to X? While it is impossible to give general $L\\sp{p}$-inequalities for $p > 1,$ we show that there are sharp weak-type inequalities, and under the additional assumption that X is bounded, sharp bounds on the distribution of the maximal function $Y\\sp\\*$ of $Y.$ For example, for all $\\lambda > 0$,$$\\lambda P(Y\\sp\\*\\ge\\lambda)\\le 6\\Vert X\\Vert\\sb1$$and the constant 6 is the best possible. In fact, if $\\beta 0,$ even the one-sided inequality $\\lambda P(\\sup\\sb{t\\ge 0}Y\\sb{t}\\ge\\lambda)>\\beta$ holds. We establish these inequalities by first giving more general inequalities for discrete-time submartingales:$$\\lambda P(g\\sp\\*\\ge\\lambda)\\le 6\\Vert f\\Vert\\sb1$$where $\\lambda > 0,\\ f = (f\\sb{n})\\sb{n\\ge 0}$ is a submartingle relative to a filtration ${\\cal F} = ({\\cal F}\\sb{n})\\sb{n\\ge 0}$, and $g = (g\\sb{n})\\sb{n\\ge 0}$ is a process also adapted to ${\\cal F}$ that is both differentially and conditionally differentially subordinate to f, i.e. with $f\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ d\\sb{k}$ and $g\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ e\\sb{k},$ we have that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ and $\\vert$E$(e\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert\\le\\vert$E$(d\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert$ for all $n\\ge 0.$ The inequalities obtained are also shown to hold for subharmonic functions and their suitably defined subordinates.","abstract_html":"Suppose X is a submartingale that is continuous on the right with limits from the left and H is a predictable process bounded by 1 in absolute value. Let $Y = (Y\\sb{t})\\sb{t\\ge 0}$ where$$Y\\sb{t} = H\\sb0X\\sb0 + \\int\\sb{(0,t\\rbrack} H\\sb{s}dX\\sb{s}.$$An interesting and important question is: How large is Y compared to X? While it is impossible to give general $L\\sp{p}$-inequalities for $p > 1,$ we show that there are sharp weak-type inequalities, and under the additional assumption that X is bounded, sharp bounds on the distribution of the maximal function $Y\\sp\\*$ of $Y.$ For example, for all $\\lambda > 0$,$$\\lambda P(Y\\sp\\*\\ge\\lambda)\\le 6\\Vert X\\Vert\\sb1$$and the constant 6 is the best possible. In fact, if $\\beta 0,$ even the one-sided inequality $\\lambda P(\\sup\\sb{t\\ge 0}Y\\sb{t}\\ge\\lambda)>\\beta$ holds. We establish these inequalities by first giving more general inequalities for discrete-time submartingales:$$\\lambda P(g\\sp\\*\\ge\\lambda)\\le 6\\Vert f\\Vert\\sb1$$where $\\lambda > 0,\\ f = (f\\sb{n})\\sb{n\\ge 0}$ is a submartingle relative to a filtration ${\\cal F} = ({\\cal F}\\sb{n})\\sb{n\\ge 0}$, and $g = (g\\sb{n})\\sb{n\\ge 0}$ is a process also adapted to ${\\cal F}$ that is both differentially and conditionally differentially subordinate to f, i.e. with $f\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ d\\sb{k}$ and $g\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ e\\sb{k},$ we have that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ and $\\vert$E$(e\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert\\le\\vert$E$(d\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert$ for all $n\\ge 0.$ The inequalities obtained are also shown to hold for subharmonic functions and their suitably defined subordinates.","abstract_has_math":true,"creators":["Hammack, William"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Peck, N.T."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:09:14Z","date_published":"2011-05-07T12:09:14Z","updated_at":"2026-07-22T22:25:14Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Hammack, William"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503205","(UMI)AAI9503205"],"render_values":[{"text":"AAI9503205","href":null,"code":true},{"text":"(UMI)AAI9503205","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19493","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Peck, N.T."]},{"key":"dc:creator","label":"Author","values":["Hammack, William"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:09:14Z","10000-01-01","1994"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Hammack, William"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503205","(UMI)AAI9503205","http://hdl.handle.net/2142/19493"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Suppose X is a submartingale that is continuous on the right with limits from the left and H is a predictable process bounded by 1 in absolute value. Let $Y = (Y\\sb{t})\\sb{t\\ge 0}$ where$$Y\\sb{t} = H\\sb0X\\sb0 + \\int\\sb{(0,t\\rbrack} H\\sb{s}dX\\sb{s}.$$An interesting and important question is: How large is Y compared to X? While it is impossible to give general $L\\sp{p}$-inequalities for $p > 1,$ we show that there are sharp weak-type inequalities, and under the additional assumption that X is bounded, sharp bounds on the distribution of the maximal function $Y\\sp\\*$ of $Y.$ For example, for all $\\lambda > 0$,$$\\lambda P(Y\\sp\\*\\ge\\lambda)\\le 6\\Vert X\\Vert\\sb1$$and the constant 6 is the best possible. In fact, if $\\beta 0,$ even the one-sided inequality $\\lambda P(\\sup\\sb{t\\ge 0}Y\\sb{t}\\ge\\lambda)>\\beta$ holds. We establish these inequalities by first giving more general inequalities for discrete-time submartingales:$$\\lambda P(g\\sp\\*\\ge\\lambda)\\le 6\\Vert f\\Vert\\sb1$$where $\\lambda > 0,\\ f = (f\\sb{n})\\sb{n\\ge 0}$ is a submartingle relative to a filtration ${\\cal F} = ({\\cal F}\\sb{n})\\sb{n\\ge 0}$, and $g = (g\\sb{n})\\sb{n\\ge 0}$ is a process also adapted to ${\\cal F}$ that is both differentially and conditionally differentially subordinate to f, i.e. with $f\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ d\\sb{k}$ and $g\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ e\\sb{k},$ we have that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ and $\\vert$E$(e\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert\\le\\vert$E$(d\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert$ for all $n\\ge 0.$ The inequalities obtained are also shown to hold for subharmonic functions and their suitably defined subordinates.","Made available in DSpace on 2011-05-07T12:09:14Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9503205.pdf: 1581932 bytes, checksum: 5b9b68601e2d060bf4455fdebafdf8cc (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:37:24Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Bounds on the size of strong subordinates of submartingales and subharmonic functions"]}]}],"canonical_facts":{"dc:contributor":["Peck, N.T."],"dc:creator":["Hammack, William"],"dc:date":["2011-05-07T12:09:14Z","10000-01-01","1994"],"dc:description":["Suppose X is a submartingale that is continuous on the right with limits from the left and H is a predictable process bounded by 1 in absolute value. Let $Y = (Y\\sb{t})\\sb{t\\ge 0}$ where$$Y\\sb{t} = H\\sb0X\\sb0 + \\int\\sb{(0,t\\rbrack} H\\sb{s}dX\\sb{s}.$$An interesting and important question is: How large is Y compared to X? While it is impossible to give general $L\\sp{p}$-inequalities for $p > 1,$ we show that there are sharp weak-type inequalities, and under the additional assumption that X is bounded, sharp bounds on the distribution of the maximal function $Y\\sp\\*$ of $Y.$ For example, for all $\\lambda > 0$,$$\\lambda P(Y\\sp\\*\\ge\\lambda)\\le 6\\Vert X\\Vert\\sb1$$and the constant 6 is the best possible. In fact, if $\\beta 0,$ even the one-sided inequality $\\lambda P(\\sup\\sb{t\\ge 0}Y\\sb{t}\\ge\\lambda)>\\beta$ holds. We establish these inequalities by first giving more general inequalities for discrete-time submartingales:$$\\lambda P(g\\sp\\*\\ge\\lambda)\\le 6\\Vert f\\Vert\\sb1$$where $\\lambda > 0,\\ f = (f\\sb{n})\\sb{n\\ge 0}$ is a submartingle relative to a filtration ${\\cal F} = ({\\cal F}\\sb{n})\\sb{n\\ge 0}$, and $g = (g\\sb{n})\\sb{n\\ge 0}$ is a process also adapted to ${\\cal F}$ that is both differentially and conditionally differentially subordinate to f, i.e. with $f\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ d\\sb{k}$ and $g\\sb{n} = {\\sum\\sbsp{k=0}{n}}\\ e\\sb{k},$ we have that $\\vert e\\sb{n}\\vert\\le\\vert d\\sb{n}\\vert$ and $\\vert$E$(e\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert\\le\\vert$E$(d\\sb{n+1}\\vert{\\cal F}\\sb{n})\\vert$ for all $n\\ge 0.$ The inequalities obtained are also shown to hold for subharmonic functions and their suitably defined subordinates.","Made available in DSpace on 2011-05-07T12:09:14Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9503205.pdf: 1581932 bytes, checksum: 5b9b68601e2d060bf4455fdebafdf8cc (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:37:24Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:15:22-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9503205","(UMI)AAI9503205","http://hdl.handle.net/2142/19493"],"dc:language":["eng"],"dc:rights":["Copyright 1994 Hammack, William"],"dc:subject":["Mathematics"],"dc:title":["Bounds on the size of strong subordinates of submartingales and subharmonic functions"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:14Z"}