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href="/wiki/Theorem" title="Theorem">theorem</a> in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, similar to <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law">Kolmogorov's zero–one law</a> and the <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli lemma</a>, that specifies that a certain type of event will either <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> happen or almost surely not happen. It is sometimes known as the <b>Savage-Hewitt law for symmetric events</b>. It is named after <a href="/wiki/Edwin_Hewitt" title="Edwin Hewitt">Edwin Hewitt</a> and <a href="/wiki/Leonard_Jimmie_Savage" title="Leonard Jimmie Savage">Leonard Jimmie Savage</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement_of_the_Hewitt-Savage_zero-one_law">Statement of the Hewitt-Savage zero-one law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hewitt%E2%80%93Savage_zero%E2%80%93one_law&action=edit&section=1" title="Edit section: Statement of the Hewitt-Savage zero-one law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a59fc3d5e5adafd7197084ff5b6d93fcdc89d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.787ex; height:3.176ex;" alt="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"></span> be a <a href="/wiki/Sequence" title="Sequence">sequence</a> of <a href="/wiki/Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">independent and identically-distributed random variables</a> taking values in a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c0954f67c8e841542d7ac6a8472c4ea739824b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {X} }"></span>. The Hewitt-Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite <a href="/wiki/Permutation" title="Permutation">permutations</a> of the indices, has <a href="/wiki/Probability" title="Probability">probability</a> either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed). </p><p>Somewhat more abstractly, define the <a href="/wiki/Invariant_sigma-algebra#Exchangeable_sigma_algebra" title="Invariant sigma-algebra">exchangeable sigma algebra</a> or <i>sigma algebra of symmetric events</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"></span> to be the set of events (depending on the sequence of variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a59fc3d5e5adafd7197084ff5b6d93fcdc89d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.787ex; height:3.176ex;" alt="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"></span>) which are invariant under <a href="https://en.wiktionary.org/wiki/finite" class="extiw" title="wikt:finite">finite</a> <a href="/wiki/Permutation" title="Permutation">permutations</a> of the indices in the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a59fc3d5e5adafd7197084ff5b6d93fcdc89d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.787ex; height:3.176ex;" alt="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in {\mathcal {E}}\implies \mathbb {P} (A)\in \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in {\mathcal {E}}\implies \mathbb {P} (A)\in \{0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68cfbba52bd6c12e92c240ea1832996a948d3a1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.778ex; height:2.843ex;" alt="{\displaystyle A\in {\mathcal {E}}\implies \mathbb {P} (A)\in \{0,1\}}"></span>. </p><p>Since any finite permutation can be written as a product of <a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">transpositions</a>, if we wish to check whether or not an event <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is symmetric (lies in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"></span>), it is enough to check if its occurrence is unchanged by an arbitrary transposition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bce188cec0088adf12527c02b5f36d13dbe5502" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.313ex; height:2.509ex;" alt="{\displaystyle i,j\in \mathbb {N} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hewitt%E2%80%93Savage_zero%E2%80%93one_law&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example_1">Example 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hewitt%E2%80%93Savage_zero%E2%80%93one_law&action=edit&section=3" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let the sequence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a59fc3d5e5adafd7197084ff5b6d93fcdc89d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.787ex; height:3.176ex;" alt="{\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}"></span> of independent and identically distributed random variables take values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}"></span>. Then the event that the series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86defb08dc5b690f35b4de3bd680941f63bd2dfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.885ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }X_{n}}"></span> converges (to a finite value) is a symmetric event in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"></span>, since its occurrence is unchanged under transpositions (for a finite re-ordering, the convergence or divergence of the series—and, indeed, the numerical value of the sum itself—is independent of the order in which we add up the terms). Thus, the series either converges almost surely or diverges almost surely. If we assume in addition that the common <a href="/wiki/Expected_value" title="Expected value">expected value</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [X_{n}]>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{n}]>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa32fd8af42f6475b96bef64a96940382d8a35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.248ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{n}]>0}"></span> (which essentially means that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (X_{n}=0)<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X_{n}=0)<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107b4080016f27c4073c6624f99e2ee74b2d3315" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.894ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (X_{n}=0)<1}"></span> because of the random variables' non-negativity), we may conclude that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} \left(\sum _{n=1}^{\infty }X_{n}=+\infty \right)=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} \left(\sum _{n=1}^{\infty }X_{n}=+\infty \right)=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0504040f3ea35fd74a901ef269f144aeb74c3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.512ex; height:7.509ex;" alt="{\displaystyle \mathbb {P} \left(\sum _{n=1}^{\infty }X_{n}=+\infty \right)=1,}"></span></dd></dl> <p>i.e. the series diverges almost surely. This is a particularly simple application of the Hewitt–Savage zero–one law. In many situations, it can be easy to apply the Hewitt–Savage zero–one law to show that some event has probability 0 or 1, but surprisingly hard to determine <i>which</i> of these two extreme values is the correct one. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example 2</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hewitt%E2%80%93Savage_zero%E2%80%93one_law&action=edit&section=4" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Continuing with the previous example, define </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0e68b1c5d20b9fcf0fcd0545720c8c45579974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.746ex; height:7.343ex;" alt="{\displaystyle S_{N}=\sum _{n=1}^{N}X_{n},}"></span></dd></dl> <p>which is the position at step <i>N</i> of a <a href="/wiki/Random_walk" title="Random walk">random walk</a> with the <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> increments <i>X</i><sub><i>n</i></sub>. The event { <i>S</i><sub><i>N</i></sub> = 0 infinitely often } is invariant under finite permutations. Therefore, the zero–one law is applicable and one infers that the probability of a random walk with real iid increments visiting the origin infinitely often is either one or zero. Visiting the origin infinitely often is a tail event with respect to the sequence (<i>S</i><sub><i>N</i></sub>), but <i>S</i><sub><i>N</i></sub> are not independent and therefore the <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law">Kolmogorov's zero–one law</a> is not directly applicable here.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hewitt%E2%80%93Savage_zero%E2%80%93one_law&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHewittSavage1955" class="citation journal cs1"><a href="/wiki/Edwin_Hewitt" title="Edwin Hewitt">Hewitt, E.</a>; <a href="/wiki/Leonard_Jimmie_Savage" title="Leonard Jimmie Savage">Savage, L. J.</a> (1955). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1955-0076206-8">"Symmetric measures on Cartesian products"</a>. <i>Trans. Amer. Math. Soc</i>. <b>80</b>: 470–501. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1955-0076206-8">10.1090/s0002-9947-1955-0076206-8</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=Symmetric+measures+on+Cartesian+products&rft.volume=80&rft.pages=470-501&rft.date=1955&rft_id=info%3Adoi%2F10.1090%2Fs0002-9947-1955-0076206-8&rft.aulast=Hewitt&rft.aufirst=E.&rft.au=Savage%2C+L.+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252Fs0002-9947-1955-0076206-8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHewitt%E2%80%93Savage+zero%E2%80%93one+law" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">This example is from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShiryaev1996" class="citation book cs1"><a href="/wiki/Albert_Shiryaev" title="Albert Shiryaev">Shiryaev, A.</a> (1996). <i>Probability Theory</i> (Second ed.). New York: Springer-Verlag. pp. 381–82.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+Theory&rft.place=New+York&rft.pages=381-82&rft.edition=Second&rft.pub=Springer-Verlag&rft.date=1996&rft.aulast=Shiryaev&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHewitt%E2%80%93Savage+zero%E2%80%93one+law" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hewitt–Savage_zero–one_law&oldid=1208880318">https://en.wikipedia.org/w/index.php?title=Hewitt–Savage_zero–one_law&oldid=1208880318</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Probability_theorems" title="Category:Probability theorems">Probability theorems</a></li><li><a href="/wiki/Category:Covering_lemmas" title="Category:Covering lemmas">Covering lemmas</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 19 February 2024, at 09:34<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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