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# Chebyshev–Markov–Stieltjes inequalities

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 Title: Chebyshev–Markov–Stieltjes inequalities Author: World Heritage Encyclopedia Language: English Subject: Andrey Markov Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Chebyshev–Markov–Stieltjes inequalities

In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes.[1] Informally, they provide sharp bounds on a measure from above and from below in terms of its first moments.

## Formulation

Given m0,...,m2m-1R, consider the collection C of measures μ on R such that

\int x^k d\mu(x) = m_k

for k = 0,1,...,2m − 1 (and in particular the integral is defined and finite).

Let P0,P1, ...,Pm be the first m + 1 orthogonal polynomials with respect to μC, and let ξ1,...ξm be the zeros of Pm. It is not hard to see that the polynomials P0,P1, ...,Pm-1 and the numbers ξ1,...ξm are the same for every μC, and therefore are determined uniquely by m0,...,m2m-1.

Denote

\rho_{m-1}(z) = 1 \Big/ \sum_{k=0}^{m-1} |P_k(z)|^2.

Theorem For j = 1,2,...,m, and any μC,

\mu(-\infty, \xi_j] \leq \rho_{m-1}(\xi_1) + \cdots + \rho_{m-1}(\xi_j) \leq \mu(-\infty,\xi_{j+1}).

## References

-- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p-------------------------------------------------------------------------------- -- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p.formatLink(frame) local args = getArgs(frame) local link = args[1] local display = args[2] if not link then return p.makeWikitextError( 'no link specified', 'Template:Format hatnote link#Errors', args.category ) end return p._formatLink(link, display) end

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p
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