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Snapshot Format
Joe Corneli edited this page Jun 6, 2013
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The current format produced by the planetmath_versions module is as follows - there are various ways in which this could be improved. One of the important accompanying technical features will be a way to make and download "snapshots" in an incremental fashion.
<?xml version="1.0" encoding="UTF-8"?>
<record version="9" id="33687">
<type>Theorem</type><comment>trigger rebuild</comment><title>Kolmogorov's inequality</title><name>KolmogorovsInequality</name><created>1039349806</created><modified>1363952541</modified><preamble>% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here</preamble><content>Let $X_1,\dots, X_n$ be independent random variables in a probability space, such that $\operatorname{E}[X_k]=0$ and $\operatorname{Var}[X_k] <\infty$ for
$k=1,\dots, n$. Then, for each $\lambda>0$,
\[P\left(\max_{1\leq k\leq n} |S_k|\geq\lambda\right)\leq \frac{1}{\lambda^2}\operatorname{Var}[S_n] = \frac{1}{\lambda^2}\sum_{k=1}^n \operatorname{Var}[X_k], \]
where $S_k = X_1 +\cdots + X_k$.
%%%%%
%%%%%</content><section class="ltx_document" prefix="xsd: http://www.w3.org/2001/XMLSchema# pmuser: http://planetmath.org/ msc: http://msc2010.org/resources/MSC/2010/ pm: http://planetmath.org/ns/ dct: http://purl.org/dc/terms/ pmarticle: http://planetmath.org/">
<hgroup><h1 class="ltx_title ltx_title_document">Kolmogorov’s inequality</h1>
</hgroup><div class="ltx_rdf" property="dct:identifier" content="KolmogorovsInequality"/>
<div class="ltx_rdf" property="dct:created" datatype="xsd:date" content="2013-03-21 15:13:03"/>
<div class="ltx_rdf" property="dct:modified" datatype="xsd:date" content="2013-03-21 15:13:03"/>
<div class="ltx_rdf" resource="pmuser:Koro" property="pm:owner"/>
<div class="ltx_rdf" resource="pmuser:Koro" property="pm:modifier"/>
<div class="ltx_rdf" property="dct:title" content="Kolmogorov's inequality"/>
<div class="ltx_rdf" property="dct:hasVersion" content="8"/>
<div class="ltx_rdf" property="pm:privacy" datatype="xsd:integer" content="1"/>
<div class="ltx_rdf" resource="pmuser:Koro" property="dct:creator"/>
<div class="ltx_rdf" property="dct:type" content="Theorem"/>
<div class="ltx_rdf" resource="msc:60E15" property="dct:subject"/>
<div class="ltx_rdf" resource="pmarticle:ChebyshevsInequality2" property="pm:related"/>
<div class="ltx_rdf" resource="pmarticle:MarkovsInequality" property="pm:related"/>
<div class="ltx_rdf" resource="pmarticle:ChebyshevsInequality" property="pm:related"/>
<div id="p1" class="ltx_para">
<p id="p1.1" class="ltx_p">Let <math id="p1.1.m1.1" class="ltx_Math" alttext="X_{1},\dots,X_{n}" display="inline" xref="p1.1.m1.1.cmml"><semantics id="p1.1.m1.1a" xref="p1.1.m1.1.cmml"><mrow id="p1.1.m1.1.8" xref="p1.1.m1.1.8.cmml"><msub id="p1.1.m1.1.8.2" xref="p1.1.m1.1.8.2.cmml"><mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">X</mi><mn id="p1.1.m1.1.2.1" xref="p1.1.m1.1.2.1.cmml">1</mn></msub><mo id="p1.1.m1.1.8a" xref="p1.1.m1.1.8.cmml">,</mo><mi mathvariant="normal" id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml">…</mi><mo id="p1.1.m1.1.8b" xref="p1.1.m1.1.8.cmml">,</mo><msub id="p1.1.m1.1.8.3" xref="p1.1.m1.1.8.3.cmml"><mi id="p1.1.m1.1.6" xref="p1.1.m1.1.6.cmml">X</mi><mi id="p1.1.m1.1.7.1" xref="p1.1.m1.1.7.1.cmml">n</mi></msub></mrow><annotation-xml id="p1.1.m1.1.cmml" encoding="MathML-Content" xref="p1.1.m1.1"><apply id="p1.1.m1.1.8.cmml" xref="p1.1.m1.1.8"><list id="p1.1.m1.1.8.1.cmml"/><apply id="p1.1.m1.1.8.2.cmml" xref="p1.1.m1.1.8.2"><csymbol cd="ambiguous" id="p1.1.m1.1.8.2.1.cmml">subscript</csymbol><ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">X</ci><cn type="integer" id="p1.1.m1.1.2.1.cmml" xref="p1.1.m1.1.2.1">1</cn></apply><ci id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4">normal-…</ci><apply id="p1.1.m1.1.8.3.cmml" xref="p1.1.m1.1.8.3"><csymbol cd="ambiguous" id="p1.1.m1.1.8.3.1.cmml">subscript</csymbol><ci id="p1.1.m1.1.6.cmml" xref="p1.1.m1.1.6">X</ci><ci id="p1.1.m1.1.7.1.cmml" xref="p1.1.m1.1.7.1">n</ci></apply></apply></annotation-xml><annotation id="p1.1.m1.1b" encoding="application/x-tex" xref="p1.1.m1.1.cmml">X_{1},\dots,X_{n}</annotation></semantics></math> be <a property="http://planetmath.org/ns/linksTo" href="http://planetmath.org/node/31053">independent</a> <a property="http://planetmath.org/ns/linksTo" href="http://planetmath.org/node/30485">random variables</a> in a <a property="http://planetmath.org/ns/linksTo" href="http://planetmath.org/node/30756">probability space</a>, such that <math id="p1.1.m2.1" class="ltx_Math" alttext="\operatorname{E}[X_{k}]=0" display="inline" xref="p1.1.m2.1.cmml"><semantics id="p1.1.m2.1a" xref="p1.1.m2.1.cmml"><mrow id="p1.1.m2.1.8" xref="p1.1.m2.1.8.cmml"><mrow id="p1.1.m2.1.8.1" xref="p1.1.m2.1.8.1.cmml"><mo id="p1.1.m2.1.1a" xref="p1.1.m2.1.1a.cmml">E</mo><mo id="p1.1.m2.1.8.1a" xref="p1.1.m2.1.8.1.cmml"></mo><mrow id="p1.1.m2.1.8.1b" xref="p1.1.m2.1.8.1.cmml"><mo id="p1.1.m2.1.8.1c" xref="p1.1.m2.1.8.1.cmml">[</mo><msub id="p1.1.m2.1.8.1.1" xref="p1.1.m2.1.8.1.1.cmml"><mi id="p1.1.m2.1.3" xref="p1.1.m2.1.3.cmml">X</mi><mi id="p1.1.m2.1.4.1" xref="p1.1.m2.1.4.1.cmml">k</mi></msub><mo id="p1.1.m2.1.8.1d" xref="p1.1.m2.1.8.1.cmml">]</mo></mrow></mrow><mo id="p1.1.m2.1.6" xref="p1.1.m2.1.6.cmml">=</mo><mn id="p1.1.m2.1.7" xref="p1.1.m2.1.7.cmml">0</mn></mrow><annotation-xml id="p1.1.m2.1.cmml" encoding="MathML-Content" xref="p1.1.m2.1"><apply id="p1.1.m2.1.8.cmml" xref="p1.1.m2.1.8"><eq id="p1.1.m2.1.6.cmml" xref="p1.1.m2.1.6"/><apply id="p1.1.m2.1.8.1.cmml" xref="p1.1.m2.1.8.1"><ci id="p1.1.m2.1.1a.cmml" xref="p1.1.m2.1.1a">normal-E</ci><apply id="p1.1.m2.1.8.1.1.cmml" xref="p1.1.m2.1.8.1.1"><csymbol cd="ambiguous" id="p1.1.m2.1.8.1.1.1.cmml">subscript</csymbol><ci id="p1.1.m2.1.3.cmml" xref="p1.1.m2.1.3">X</ci><ci id="p1.1.m2.1.4.1.cmml" xref="p1.1.m2.1.4.1">k</ci></apply></apply><cn type="integer" id="p1.1.m2.1.7.cmml" xref="p1.1.m2.1.7">0</cn></apply></annotation-xml><annotation id="p1.1.m2.1b" encoding="application/x-tex" xref="p1.1.m2.1.cmml">\operatorname{E}[X_{k}]=0</annotation></semantics></math> and <math id="p1.1.m3.1" class="ltx_Math" alttext="\operatorname{Var}[X_{k}]<\infty" display="inline" xref="p1.1.m3.1.cmml"><semantics id="p1.1.m3.1a" xref="p1.1.m3.1.cmml"><mrow id="p1.1.m3.1.8" xref="p1.1.m3.1.8.cmml"><mrow id="p1.1.m3.1.8.1" xref="p1.1.m3.1.8.1.cmml"><mo id="p1.1.m3.1.1a" xref="p1.1.m3.1.1a.cmml">Var</mo><mo id="p1.1.m3.1.8.1a" xref="p1.1.m3.1.8.1.cmml"></mo><mrow id="p1.1.m3.1.8.1b" xref="p1.1.m3.1.8.1.cmml"><mo id="p1.1.m3.1.8.1c" xref="p1.1.m3.1.8.1.cmml">[</mo><msub id="p1.1.m3.1.8.1.1" xref="p1.1.m3.1.8.1.1.cmml"><mi id="p1.1.m3.1.3" xref="p1.1.m3.1.3.cmml">X</mi><mi id="p1.1.m3.1.4.1" xref="p1.1.m3.1.4.1.cmml">k</mi></msub><mo id="p1.1.m3.1.8.1d" xref="p1.1.m3.1.8.1.cmml">]</mo></mrow></mrow><mo id="p1.1.m3.1.6" xref="p1.1.m3.1.6.cmml"><</mo><mi mathvariant="normal" id="p1.1.m3.1.7" xref="p1.1.m3.1.7.cmml">∞</mi></mrow><annotation-xml id="p1.1.m3.1.cmml" encoding="MathML-Content" xref="p1.1.m3.1"><apply id="p1.1.m3.1.8.cmml" xref="p1.1.m3.1.8"><lt id="p1.1.m3.1.6.cmml" xref="p1.1.m3.1.6"/><apply id="p1.1.m3.1.8.1.cmml" xref="p1.1.m3.1.8.1"><ci id="p1.1.m3.1.1a.cmml" xref="p1.1.m3.1.1a">Var</ci><apply id="p1.1.m3.1.8.1.1.cmml" xref="p1.1.m3.1.8.1.1"><csymbol cd="ambiguous" id="p1.1.m3.1.8.1.1.1.cmml">subscript</csymbol><ci id="p1.1.m3.1.3.cmml" xref="p1.1.m3.1.3">X</ci><ci id="p1.1.m3.1.4.1.cmml" xref="p1.1.m3.1.4.1">k</ci></apply></apply><infinity id="p1.1.m3.1.7.cmml" xref="p1.1.m3.1.7"/></apply></annotation-xml><annotation id="p1.1.m3.1b" encoding="application/x-tex" xref="p1.1.m3.1.cmml">\operatorname{Var}[X_{k}]<\infty</annotation></semantics></math> for
<math id="p1.1.m4.1" class="ltx_Math" alttext="k=1,\dots,n" display="inline" xref="p1.1.m4.1.cmml"><semantics id="p1.1.m4.1a" xref="p1.1.m4.1.cmml"><mrow id="p1.1.m4.1.8" xref="p1.1.m4.1.8.cmml"><mi id="p1.1.m4.1.1" xref="p1.1.m4.1.1.cmml">k</mi><mo id="p1.1.m4.1.2" xref="p1.1.m4.1.2.cmml">=</mo><mrow id="p1.1.m4.1.8.1" xref="p1.1.m4.1.8.1.cmml"><mn id="p1.1.m4.1.3" xref="p1.1.m4.1.3.cmml">1</mn><mo id="p1.1.m4.1.8.1a" xref="p1.1.m4.1.8.1.cmml">,</mo><mi mathvariant="normal" id="p1.1.m4.1.5" xref="p1.1.m4.1.5.cmml">…</mi><mo id="p1.1.m4.1.8.1b" xref="p1.1.m4.1.8.1.cmml">,</mo><mi id="p1.1.m4.1.7" xref="p1.1.m4.1.7.cmml">n</mi></mrow></mrow><annotation-xml id="p1.1.m4.1.cmml" encoding="MathML-Content" xref="p1.1.m4.1"><apply id="p1.1.m4.1.8.cmml" xref="p1.1.m4.1.8"><eq id="p1.1.m4.1.2.cmml" xref="p1.1.m4.1.2"/><ci id="p1.1.m4.1.1.cmml" xref="p1.1.m4.1.1">k</ci><apply id="p1.1.m4.1.8.1.cmml" xref="p1.1.m4.1.8.1"><list id="p1.1.m4.1.8.1.1.cmml"/><cn type="integer" id="p1.1.m4.1.3.cmml" xref="p1.1.m4.1.3">1</cn><ci id="p1.1.m4.1.5.cmml" xref="p1.1.m4.1.5">normal-…</ci><ci id="p1.1.m4.1.7.cmml" xref="p1.1.m4.1.7">n</ci></apply></apply></annotation-xml><annotation id="p1.1.m4.1b" encoding="application/x-tex" xref="p1.1.m4.1.cmml">k=1,\dots,n</annotation></semantics></math>. Then, for each <math id="p1.1.m5.1" class="ltx_Math" alttext="\lambda>0" display="inline" xref="p1.1.m5.1.cmml"><semantics id="p1.1.m5.1a" xref="p1.1.m5.1.cmml"><mrow id="p1.1.m5.1.4" xref="p1.1.m5.1.4.cmml"><mi id="p1.1.m5.1.1" xref="p1.1.m5.1.1.cmml">λ</mi><mo id="p1.1.m5.1.2" xref="p1.1.m5.1.2.cmml">></mo><mn id="p1.1.m5.1.3" xref="p1.1.m5.1.3.cmml">0</mn></mrow><annotation-xml id="p1.1.m5.1.cmml" encoding="MathML-Content" xref="p1.1.m5.1"><apply id="p1.1.m5.1.4.cmml" xref="p1.1.m5.1.4"><gt id="p1.1.m5.1.2.cmml" xref="p1.1.m5.1.2"/><ci id="p1.1.m5.1.1.cmml" xref="p1.1.m5.1.1">λ</ci><cn type="integer" id="p1.1.m5.1.3.cmml" xref="p1.1.m5.1.3">0</cn></apply></annotation-xml><annotation id="p1.1.m5.1b" encoding="application/x-tex" xref="p1.1.m5.1.cmml">\lambda>0</annotation></semantics></math>,</p>
</div>
<div id="p2" class="ltx_para">
<table id="S0.Ex1" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math id="S0.Ex1.m1.1" class="ltx_Math" alttext="P\left(\max_{{1\leq k\leq n}}|S_{k}|\geq\lambda\right)\leq\frac{1}{\lambda^{2}% }\operatorname{Var}[S_{n}]=\frac{1}{\lambda^{2}}\sum_{{k=1}}^{n}\operatorname{% Var}[X_{k}]," display="block" xref="S0.Ex1.m1.1.cmml"><semantics id="S0.Ex1.m1.1a" xref="S0.Ex1.m1.1.cmml"><mrow id="S0.Ex1.m1.1b" xref="S0.Ex1.m1.1.cmml"><mi id="S0.Ex1.m1.1.1">P</mi><mrow id="S0.Ex1.m1.1.30" xref="S0.Ex1.m1.1.30.cmml"><mo id="S0.Ex1.m1.1.2" xref="S0.Ex1.m1.1.2.cmml">(</mo><munder id="S0.Ex1.m1.1.30.1" xref="S0.Ex1.m1.1.30.1.cmml"><mo movablelimits="false" id="S0.Ex1.m1.1.3" xref="S0.Ex1.m1.1.3.cmml">max</mo><mrow id="S0.Ex1.m1.1.4.1" xref="S0.Ex1.m1.1.4.1.cmml"><mn id="S0.Ex1.m1.1.4.1.1" xref="S0.Ex1.m1.1.4.1.1.cmml">1</mn><mo id="S0.Ex1.m1.1.4.1.2" xref="S0.Ex1.m1.1.4.1.2.cmml">≤</mo><mi id="S0.Ex1.m1.1.4.1.3" xref="S0.Ex1.m1.1.4.1.3.cmml">k</mi><mo id="S0.Ex1.m1.1.4.1.4" xref="S0.Ex1.m1.1.4.1.4.cmml">≤</mo><mi id="S0.Ex1.m1.1.4.1.5" xref="S0.Ex1.m1.1.4.1.5.cmml">n</mi></mrow></munder><mo id="S0.Ex1.m1.1.5" xref="S0.Ex1.m1.1.5.cmml">|</mo><msub id="S0.Ex1.m1.1.30.2" xref="S0.Ex1.m1.1.30.2.cmml"><mi id="S0.Ex1.m1.1.6" xref="S0.Ex1.m1.1.6.cmml">S</mi><mi id="S0.Ex1.m1.1.7.1" xref="S0.Ex1.m1.1.7.1.cmml">k</mi></msub><mo id="S0.Ex1.m1.1.8" xref="S0.Ex1.m1.1.8.cmml">|</mo><mo id="S0.Ex1.m1.1.9" xref="S0.Ex1.m1.1.9.cmml">≥</mo><mi id="S0.Ex1.m1.1.10">λ</mi><mo id="S0.Ex1.m1.1.11" xref="S0.Ex1.m1.1.11.cmml">)</mo></mrow><mo id="S0.Ex1.m1.1.12" xref="S0.Ex1.m1.1.12.cmml">≤</mo><mfrac id="S0.Ex1.m1.1.13" xref="S0.Ex1.m1.1.13.cmml"><mn id="S0.Ex1.m1.1.13.2" xref="S0.Ex1.m1.1.13.2.cmml">1</mn><msup id="S0.Ex1.m1.1.13.3" xref="S0.Ex1.m1.1.13.3.cmml"><mi id="S0.Ex1.m1.1.13.3.1" xref="S0.Ex1.m1.1.13.3.1.cmml">λ</mi><mn id="S0.Ex1.m1.1.13.3.2.1" xref="S0.Ex1.m1.1.13.3.2.1.cmml">2</mn></msup></mfrac><mo id="S0.Ex1.m1.1.14a" xref="S0.Ex1.m1.1.14a.cmml">Var</mo><mrow id="S0.Ex1.m1.1.31" xref="S0.Ex1.m1.1.31.cmml"><mo id="S0.Ex1.m1.1.15" xref="S0.Ex1.m1.1.15.cmml">[</mo><msub id="S0.Ex1.m1.1.31.1" xref="S0.Ex1.m1.1.31.1.cmml"><mi id="S0.Ex1.m1.1.16" xref="S0.Ex1.m1.1.16.cmml">S</mi><mi id="S0.Ex1.m1.1.17.1" xref="S0.Ex1.m1.1.17.1.cmml">n</mi></msub><mo id="S0.Ex1.m1.1.18" xref="S0.Ex1.m1.1.18.cmml">]</mo></mrow><mo id="S0.Ex1.m1.1.19" xref="S0.Ex1.m1.1.19.cmml">=</mo><mfrac id="S0.Ex1.m1.1.20" xref="S0.Ex1.m1.1.20.cmml"><mn id="S0.Ex1.m1.1.20.2" xref="S0.Ex1.m1.1.20.2.cmml">1</mn><msup id="S0.Ex1.m1.1.20.3" xref="S0.Ex1.m1.1.20.3.cmml"><mi id="S0.Ex1.m1.1.20.3.1" xref="S0.Ex1.m1.1.20.3.1.cmml">λ</mi><mn id="S0.Ex1.m1.1.20.3.2.1" xref="S0.Ex1.m1.1.20.3.2.1.cmml">2</mn></msup></mfrac><mover id="S0.Ex1.m1.1.32" xref="S0.Ex1.m1.1.32.cmml"><munder id="S0.Ex1.m1.1.32.2" xref="S0.Ex1.m1.1.32.2.cmml"><mo movablelimits="false" id="S0.Ex1.m1.1.21" xref="S0.Ex1.m1.1.21.cmml">∑</mo><mrow id="S0.Ex1.m1.1.22.1" xref="S0.Ex1.m1.1.22.1.cmml"><mi id="S0.Ex1.m1.1.22.1.1" xref="S0.Ex1.m1.1.22.1.1.cmml">k</mi><mo movablelimits="false" id="S0.Ex1.m1.1.22.1.2" xref="S0.Ex1.m1.1.22.1.2.cmml">=</mo><mn id="S0.Ex1.m1.1.22.1.3" xref="S0.Ex1.m1.1.22.1.3.cmml">1</mn></mrow></munder><mi id="S0.Ex1.m1.1.23.1" xref="S0.Ex1.m1.1.23.1.cmml">n</mi></mover><mo id="S0.Ex1.m1.1.24a" xref="S0.Ex1.m1.1.24a.cmml">Var</mo><mrow id="S0.Ex1.m1.1.33" xref="S0.Ex1.m1.1.33.cmml"><mo id="S0.Ex1.m1.1.25" xref="S0.Ex1.m1.1.25.cmml">[</mo><msub id="S0.Ex1.m1.1.33.1" xref="S0.Ex1.m1.1.33.1.cmml"><mi id="S0.Ex1.m1.1.26" xref="S0.Ex1.m1.1.26.cmml">X</mi><mi id="S0.Ex1.m1.1.27.1" xref="S0.Ex1.m1.1.27.1.cmml">k</mi></msub><mo id="S0.Ex1.m1.1.28" xref="S0.Ex1.m1.1.28.cmml">]</mo></mrow><mo id="S0.Ex1.m1.1.29" xref="S0.Ex1.m1.1.29.cmml">,</mo></mrow><annotation-xml id="S0.Ex1.m1.1.cmml" encoding="MathML-Content" xref="S0.Ex1.m1.1"><cerror id="S0.Ex1.m1.1a.cmml" xref="S0.Ex1.m1.1"><csymbol cd="ambiguous" id="S0.Ex1.m1.1b.cmml" xref="S0.Ex1.m1.1">fragments</csymbol><csymbol cd="unknown" id="S0.Ex1.m1.1c.cmml" xref="S0.Ex1.m1.1">P</csymbol><cerror id="S0.Ex1.m1.1.30.cmml" xref="S0.Ex1.m1.1.30"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.30a.cmml" xref="S0.Ex1.m1.1.30">fragments</csymbol><ci id="S0.Ex1.m1.1.2.cmml" xref="S0.Ex1.m1.1.2">normal-(</ci><apply id="S0.Ex1.m1.1.30.1.cmml" xref="S0.Ex1.m1.1.30.1"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.30.1.1.cmml">subscript</csymbol><max id="S0.Ex1.m1.1.3.cmml" xref="S0.Ex1.m1.1.3"/><apply id="S0.Ex1.m1.1.4.1.cmml" xref="S0.Ex1.m1.1.4.1"><and id="S0.Ex1.m1.1.4.1a.cmml" xref="S0.Ex1.m1.1.4.1"/><apply id="S0.Ex1.m1.1.4.1b.cmml" xref="S0.Ex1.m1.1.4.1"><leq id="S0.Ex1.m1.1.4.1.2.cmml" xref="S0.Ex1.m1.1.4.1.2"/><cn type="integer" id="S0.Ex1.m1.1.4.1.1.cmml" xref="S0.Ex1.m1.1.4.1.1">1</cn><ci id="S0.Ex1.m1.1.4.1.3.cmml" xref="S0.Ex1.m1.1.4.1.3">k</ci></apply><apply id="S0.Ex1.m1.1.4.1c.cmml" xref="S0.Ex1.m1.1.4.1"><leq id="S0.Ex1.m1.1.4.1.4.cmml" xref="S0.Ex1.m1.1.4.1.4"/><share href="#S0.Ex1.m1.1.4.1.3.cmml" id="S0.Ex1.m1.1.4.1d.cmml" xref="S0.Ex1.m1.1.4.1"/><ci id="S0.Ex1.m1.1.4.1.5.cmml" xref="S0.Ex1.m1.1.4.1.5">n</ci></apply></apply></apply><ci id="S0.Ex1.m1.1.5.cmml" xref="S0.Ex1.m1.1.5">normal-|</ci><apply id="S0.Ex1.m1.1.30.2.cmml" xref="S0.Ex1.m1.1.30.2"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.30.2.1.cmml">subscript</csymbol><ci id="S0.Ex1.m1.1.6.cmml" xref="S0.Ex1.m1.1.6">S</ci><ci id="S0.Ex1.m1.1.7.1.cmml" xref="S0.Ex1.m1.1.7.1">k</ci></apply><ci id="S0.Ex1.m1.1.8.cmml" xref="S0.Ex1.m1.1.8">normal-|</ci><geq id="S0.Ex1.m1.1.9.cmml" xref="S0.Ex1.m1.1.9"/><csymbol cd="unknown" id="S0.Ex1.m1.1.30b.cmml" xref="S0.Ex1.m1.1.30">λ</csymbol><ci id="S0.Ex1.m1.1.11.cmml" xref="S0.Ex1.m1.1.11">normal-)</ci></cerror><leq id="S0.Ex1.m1.1.12.cmml" xref="S0.Ex1.m1.1.12"/><apply id="S0.Ex1.m1.1.13.cmml" xref="S0.Ex1.m1.1.13"><divide id="S0.Ex1.m1.1.13.1.cmml"/><cn type="integer" id="S0.Ex1.m1.1.13.2.cmml" xref="S0.Ex1.m1.1.13.2">1</cn><apply id="S0.Ex1.m1.1.13.3.cmml" xref="S0.Ex1.m1.1.13.3"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.13.3.3.cmml">superscript</csymbol><ci id="S0.Ex1.m1.1.13.3.1.cmml" xref="S0.Ex1.m1.1.13.3.1">λ</ci><cn type="integer" id="S0.Ex1.m1.1.13.3.2.1.cmml" xref="S0.Ex1.m1.1.13.3.2.1">2</cn></apply></apply><ci id="S0.Ex1.m1.1.14a.cmml" xref="S0.Ex1.m1.1.14a">Var</ci><cerror id="S0.Ex1.m1.1.31.cmml" xref="S0.Ex1.m1.1.31"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.31a.cmml" xref="S0.Ex1.m1.1.31">fragments</csymbol><ci id="S0.Ex1.m1.1.15.cmml" xref="S0.Ex1.m1.1.15">normal-[</ci><apply id="S0.Ex1.m1.1.31.1.cmml" xref="S0.Ex1.m1.1.31.1"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.31.1.1.cmml">subscript</csymbol><ci id="S0.Ex1.m1.1.16.cmml" xref="S0.Ex1.m1.1.16">S</ci><ci id="S0.Ex1.m1.1.17.1.cmml" xref="S0.Ex1.m1.1.17.1">n</ci></apply><ci id="S0.Ex1.m1.1.18.cmml" xref="S0.Ex1.m1.1.18">normal-]</ci></cerror><eq id="S0.Ex1.m1.1.19.cmml" xref="S0.Ex1.m1.1.19"/><apply id="S0.Ex1.m1.1.20.cmml" xref="S0.Ex1.m1.1.20"><divide id="S0.Ex1.m1.1.20.1.cmml"/><cn type="integer" id="S0.Ex1.m1.1.20.2.cmml" xref="S0.Ex1.m1.1.20.2">1</cn><apply id="S0.Ex1.m1.1.20.3.cmml" xref="S0.Ex1.m1.1.20.3"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.20.3.3.cmml">superscript</csymbol><ci id="S0.Ex1.m1.1.20.3.1.cmml" xref="S0.Ex1.m1.1.20.3.1">λ</ci><cn type="integer" id="S0.Ex1.m1.1.20.3.2.1.cmml" xref="S0.Ex1.m1.1.20.3.2.1">2</cn></apply></apply><apply id="S0.Ex1.m1.1.32.cmml" xref="S0.Ex1.m1.1.32"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.32.1.cmml">superscript</csymbol><apply id="S0.Ex1.m1.1.32.2.cmml" xref="S0.Ex1.m1.1.32.2"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.32.2.1.cmml">subscript</csymbol><sum id="S0.Ex1.m1.1.21.cmml" xref="S0.Ex1.m1.1.21"/><apply id="S0.Ex1.m1.1.22.1.cmml" xref="S0.Ex1.m1.1.22.1"><eq id="S0.Ex1.m1.1.22.1.2.cmml" xref="S0.Ex1.m1.1.22.1.2"/><ci id="S0.Ex1.m1.1.22.1.1.cmml" xref="S0.Ex1.m1.1.22.1.1">k</ci><cn type="integer" id="S0.Ex1.m1.1.22.1.3.cmml" xref="S0.Ex1.m1.1.22.1.3">1</cn></apply></apply><ci id="S0.Ex1.m1.1.23.1.cmml" xref="S0.Ex1.m1.1.23.1">n</ci></apply><ci id="S0.Ex1.m1.1.24a.cmml" xref="S0.Ex1.m1.1.24a">Var</ci><cerror id="S0.Ex1.m1.1.33.cmml" xref="S0.Ex1.m1.1.33"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.33a.cmml" xref="S0.Ex1.m1.1.33">fragments</csymbol><ci id="S0.Ex1.m1.1.25.cmml" xref="S0.Ex1.m1.1.25">normal-[</ci><apply id="S0.Ex1.m1.1.33.1.cmml" xref="S0.Ex1.m1.1.33.1"><csymbol cd="ambiguous" id="S0.Ex1.m1.1.33.1.1.cmml">subscript</csymbol><ci id="S0.Ex1.m1.1.26.cmml" xref="S0.Ex1.m1.1.26">X</ci><ci id="S0.Ex1.m1.1.27.1.cmml" xref="S0.Ex1.m1.1.27.1">k</ci></apply><ci id="S0.Ex1.m1.1.28.cmml" xref="S0.Ex1.m1.1.28">normal-]</ci></cerror><ci id="S0.Ex1.m1.1.29.cmml" xref="S0.Ex1.m1.1.29">normal-,</ci></cerror></annotation-xml><annotation id="S0.Ex1.m1.1c" encoding="application/x-tex" xref="S0.Ex1.m1.1.cmml">P\left(\max_{{1\leq k\leq n}}|S_{k}|\geq\lambda\right)\leq\frac{1}{\lambda^{2}%
}\operatorname{Var}[S_{n}]=\frac{1}{\lambda^{2}}\sum_{{k=1}}^{n}\operatorname{%
Var}[X_{k}],</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
</table>
</div>
<div id="p3" class="ltx_para">
<p id="p3.1" class="ltx_p">where <math id="p3.1.m1.1" class="ltx_Math" alttext="S_{k}=X_{1}+\cdots+X_{k}" display="inline" xref="p3.1.m1.1.cmml"><semantics id="p3.1.m1.1a" xref="p3.1.m1.1.cmml"><mrow id="p3.1.m1.1.11" xref="p3.1.m1.1.11.cmml"><msub id="p3.1.m1.1.11.1" xref="p3.1.m1.1.11.1.cmml"><mi id="p3.1.m1.1.1" xref="p3.1.m1.1.1.cmml">S</mi><mi id="p3.1.m1.1.2.1" xref="p3.1.m1.1.2.1.cmml">k</mi></msub><mo id="p3.1.m1.1.3" xref="p3.1.m1.1.3.cmml">=</mo><mrow id="p3.1.m1.1.11.2" xref="p3.1.m1.1.11.2.cmml"><msub id="p3.1.m1.1.11.2.1" xref="p3.1.m1.1.11.2.1.cmml"><mi id="p3.1.m1.1.4" xref="p3.1.m1.1.4.cmml">X</mi><mn id="p3.1.m1.1.5.1" xref="p3.1.m1.1.5.1.cmml">1</mn></msub><mo id="p3.1.m1.1.6" xref="p3.1.m1.1.6.cmml">+</mo><mi mathvariant="normal" id="p3.1.m1.1.7" xref="p3.1.m1.1.7.cmml">⋯</mi><mo id="p3.1.m1.1.6a" xref="p3.1.m1.1.6.cmml">+</mo><msub id="p3.1.m1.1.11.2.2" xref="p3.1.m1.1.11.2.2.cmml"><mi id="p3.1.m1.1.9" xref="p3.1.m1.1.9.cmml">X</mi><mi id="p3.1.m1.1.10.1" xref="p3.1.m1.1.10.1.cmml">k</mi></msub></mrow></mrow><annotation-xml id="p3.1.m1.1.cmml" encoding="MathML-Content" xref="p3.1.m1.1"><apply id="p3.1.m1.1.11.cmml" xref="p3.1.m1.1.11"><eq id="p3.1.m1.1.3.cmml" xref="p3.1.m1.1.3"/><apply id="p3.1.m1.1.11.1.cmml" xref="p3.1.m1.1.11.1"><csymbol cd="ambiguous" id="p3.1.m1.1.11.1.1.cmml">subscript</csymbol><ci id="p3.1.m1.1.1.cmml" xref="p3.1.m1.1.1">S</ci><ci id="p3.1.m1.1.2.1.cmml" xref="p3.1.m1.1.2.1">k</ci></apply><apply id="p3.1.m1.1.11.2.cmml" xref="p3.1.m1.1.11.2"><plus id="p3.1.m1.1.6.cmml" xref="p3.1.m1.1.6"/><apply id="p3.1.m1.1.11.2.1.cmml" xref="p3.1.m1.1.11.2.1"><csymbol cd="ambiguous" id="p3.1.m1.1.11.2.1.1.cmml">subscript</csymbol><ci id="p3.1.m1.1.4.cmml" xref="p3.1.m1.1.4">X</ci><cn type="integer" id="p3.1.m1.1.5.1.cmml" xref="p3.1.m1.1.5.1">1</cn></apply><ci id="p3.1.m1.1.7.cmml" xref="p3.1.m1.1.7">normal-⋯</ci><apply id="p3.1.m1.1.11.2.2.cmml" xref="p3.1.m1.1.11.2.2"><csymbol cd="ambiguous" id="p3.1.m1.1.11.2.2.1.cmml">subscript</csymbol><ci id="p3.1.m1.1.9.cmml" xref="p3.1.m1.1.9">X</ci><ci id="p3.1.m1.1.10.1.cmml" xref="p3.1.m1.1.10.1">k</ci></apply></apply></apply></annotation-xml><annotation id="p3.1.m1.1b" encoding="application/x-tex" xref="p3.1.m1.1.cmml">S_{k}=X_{1}+\cdots+X_{k}</annotation></semantics></math>.
</p>
</div>
</section><creator id="127" name="Koro"/><modifier id="127" name="Koro"/><classification><category scheme="msc" code="60E15"/></classification></record>