List of probabilistic proofs of non-probabilistic theorems

<h2 id="analysis">Analysis</h2>
<ul><li><a href="/facts/Normal_number/nGb5cM0y">Normal numbers</a> exist. Moreover, <a href="/facts/Computable_number/jRShztzh">computable</a> normal numbers <a href="/facts/Normal_number/nGb5cM0y">exist</a>. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the <a href="/facts/Strong_law_of_large_numbers/X3Bjcy3v">strong law of large numbers</a>); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments.</li>
<li><a href="/facts/Dvoretzky%2527s_theorem/7x5fYEAb">Dvoretzky's theorem</a> which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies.</li>
<li>The diameter of the <a href="/facts/Banach%25E2%2580%2593Mazur_compactum/K6AG58HI">Banach–Mazur compactum</a> was calculated using a probabilistic construction. No deterministic construction is known.</li>
<li>The original proof that the <a href="/facts/Hausdorff%25E2%2580%2593Young_inequality/uFlluIBg">Hausdorff–Young inequality</a> cannot be extended to 
 
 
 
 p
 >
 2
 
 
 {\displaystyle p>2}
 
 is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>The first construction of a Salem set was probabilistic.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Only in 1981 did Kaufman give a deterministic construction.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>Every continuous function on a compact interval can be uniformly approximated by polynomials, which is the <a href="/facts/Weierstrass_approximation_theorem/QB07Nwba">Weierstrass approximation theorem</a>. A <a href="/facts/Bernstein_polynomial/XkJFDDIB">probabilistic proof</a> uses the <a href="/facts/Weak_law_of_large_numbers/X3Bjcy3v">weak law of large numbers</a>. Non-probabilistic proofs were available earlier.</li>
<li>Existence of a nowhere differentiable continuous function follows easily from properties of <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a>. A <a href="/facts/Weierstrass_function/m9jU4LE5">non-probabilistic proof</a> was available earlier.</li>
<li><a href="/facts/Stirling%2527s_formula/GfxV3BUH">Stirling's formula</a> was first discovered by <a href="/facts/Abraham_de_Moivre/m9hpa6Mb">Abraham de Moivre</a> in his `<a href="/facts/The_Doctrine_of_Chances/FyF2UuqI">The Doctrine of Chances</a>' (with a constant identified later by Stirling) in order to be used in probability theory. Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<ul><li>The only bounded harmonic functions defined on the whole plane are constant functions by <a href="/facts/Harmonic_function/H6pg4dMT">Liouville's theorem</a>. A probabilistic proof via n-dimensional <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> is well known.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Non-probabilistic proofs were available earlier.</li>
<li>Non-tangential boundary values<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> of an <a href="/facts/Holomorphic_function/kVzthtEf">analytic</a> or <a href="/facts/Harmonic_function/H6pg4dMT">harmonic</a> function exist at almost all boundary points of non-tangential boundedness. This result (<a href="/facts/Ivan_Privalov/YuedVTWj">Privalov</a>'s theorem), and several results of this kind, are deduced from <a href="/facts/Doob%2527s_martingale_convergence_theorems/52N9mHcI">martingale convergence</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> Non-probabilistic proofs were available earlier.</li>
<li>The boundary Harnack principle is proved using Brownian motion<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> (see also<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>). Non-probabilistic proofs were available earlier.</li>
<li><a href="/facts/Basel_problem/2PManmlC">Euler's Basel sum</a>, 
 
 
 
 
 
 ∑
 
 n
 =
 1
 
 
 ∞
 
 
 
 
 1
 
 n
 
 2
 
 
 
 
 =
 
 
 
 π
 
 2
 
 
 6
 
 
 ,
 
 
 {\displaystyle \qquad \sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}},}
 
 can be demonstrated by considering the expected exit time of planar Brownian motion from an infinite strip. A number of other less well-known identities can be deduced in a similar manner.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li></ul>
<ul><li>The <a href="/facts/Picard_theorem/0cVTD10S">Picard theorem</a> can be proved using the winding properties of planar Brownian motion.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a></li>
<li>The fact that every <a href="/facts/Lipschitz_function/Hw20EPEU">Lipschitz function</a> on the real line is differentiable almost everywhere can be proved using <a href="/facts/Martingale_convergence/52N9mHcI">martingale convergence</a>.</li>
<li>Multidimensional Fourier inversion formula can be established by the <a href="/facts/Weak_law_of_large_numbers/X3Bjcy3v">weak law of large numbers</a> and some elementary results from complex analysis.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>Abért and <a href="/facts/Benjamin_Weiss/e8BSxcQV">Weiss</a> proved, via a probabilistic construction, that <a href="/facts/Bernoulli_shift/JA8N69xf">Bernoulli shifts</a> are weakly contained (in the sense of <a href="/facts/Alexander_S._Kechris/XUP7SToo">Kechris</a>) in any free <a href="/facts/Measure-preserving_dynamical_system/acIdlpqt">measure-preserving action</a> action of a discrete, countable group on a standard probability space.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a></li></ul>
<h2 id="combinatorics">Combinatorics</h2>
<ul><li>A number of theorems stating existence of graphs (and other discrete structures) with desired properties are proved by the <a href="/facts/Probabilistic_method/AXKXrzrr">probabilistic method</a>. Non-probabilistic proofs are available for a few of them.</li>
<li>The <a href="/facts/Maximum-minimums_identity/0WFh4GEH">maximum-minimums identity</a> admits a probabilistic proof.</li>
<li><a href="/facts/Crossing_number_inequality/XaqTQlFT">Crossing number inequality</a> which is a lower bound on the number of crossing for any drawing of a graph as a function of the number of vertices, edges the graph has.</li></ul>
<h2 id="algebra">Algebra</h2>
<ul><li>The <a href="/facts/Fundamental_theorem_of_algebra/vspflSsM">fundamental theorem of algebra</a> can be proved using two-dimensional Brownian motion.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> Non-probabilistic proofs were available earlier.</li>
<li>The <a href="/facts/Atiyah%25E2%2580%2593Singer_index_theorem/qwcSxamI">index theorem for elliptic complexes</a> is proved using probabilistic methods<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> (rather than heat equation methods). A non-probabilistic proof was available earlier.</li></ul>
<h2 id="topology-and-geometry">Topology and geometry</h2>
<ul><li>A smooth <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> is evidently two-sided, but a non-smooth (especially, fractal) boundary can be quite complicated. It was conjectured to be two-sided in the sense that the natural projection of the <a href="/facts/Martin_boundary/1KvVw6mG">Martin boundary</a><a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> to the topological boundary is at most 2 to 1 almost everywhere.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> This conjecture is proved using <a href="/facts/Wiener_process/KPh7vYd7">Brownian motion</a>, <a href="/facts/Local_time_(mathematics)/Pychb8n8">local time</a>, <a href="/facts/Stochastic_calculus/QgY91tp2">stochastic integration</a>, <a href="/facts/Coupling_(probability)/7Gp6yAqH">coupling</a>, hypercontractivity etc.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> (see also<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>). A non-probabilistic proof is found 18 years later.<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></li>
<li>The <a href="/facts/Introduction_to_systolic_geometry/ctr9D2cg">Loewner's torus inequality</a> relates the area of a <a href="/facts/Compact_surface/jMG6G6Bl">compact surface</a> (topologically, a torus) to its <a href="/facts/Introduction_to_systolic_geometry/ctr9D2cg">systole</a>. It can be <a href="/facts/Loewner%2527s_torus_inequality/pkF4mAxv">proved most easily</a> by using the probabilistic notion of <a href="/facts/Variance/ULBJKXD1">variance</a>.<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a> A non-probabilistic proof was available earlier.</li>
<li>The weak halfspace theorem for <a href="/facts/Minimal_surface/Fy7VkrLQ">minimal surfaces</a> states that any complete minimal surface of bounded curvature which is not a plane is not contained in any halfspace. This theorem is proved using a <a href="/facts/Coupling/JDHEKQYn">coupling</a> between Brownian motions on minimal surfaces.<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> A non-probabilistic proof was available earlier.</li></ul>
<h2 id="number-theory">Number theory</h2>
<ul><li>The <a href="/facts/Normal_number/nGb5cM0y">normal number</a> theorem (1909), due to <a href="/facts/%25C3%2589mile_Borel/lF9PBa1T">Émile Borel</a>, could be one of the first examples of the <a href="/facts/Probabilistic_method/AXKXrzrr">probabilistic method</a>, providing the first proof of existence of normal numbers, with the help of the first version of the strong <a href="/facts/Law_of_large_numbers/X3Bjcy3v">law of large numbers</a> (see also the first item of the section Analysis).</li>
<li>The <a href="/facts/Rogers%25E2%2580%2593Ramanujan_identities/TyRnK1Ql">Rogers–Ramanujan identities</a> are proved using <a href="/facts/Markov_chain/5oP5hntk">Markov chains</a>.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> A non-probabilistic proof was available earlier.</li></ul>
<h2 id="quantum-theory">Quantum theory</h2>
<ul><li>Non-commutative dynamics (called also quantum dynamics) is formulated in terms of <a href="/facts/Von_Neumann_algebra/8NyWUIRH">Von Neumann algebras</a> and continuous <a href="/facts/Tensor_product_of_Hilbert_spaces/GSavLJfU">tensor products of Hilbert spaces</a>.<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> Several results (for example, a continuum of mutually non-isomorphic models) are obtained by probabilistic means (<a href="/facts/Random_compact_set/K7UJWzqR">random compact sets</a> and <a href="/facts/Wiener_process/KPh7vYd7">Brownian motion</a>).<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a><a class="footnote-ref" id="fnref:28" href="#fn:28">28</a> One part of this theory (so-called type III systems) is translated into the analytic language<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> and is developing analytically;<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a> the other part (so-called type II systems) exists still in the probabilistic language only.</li>
<li>Tripartite quantum states can lead to arbitrary large violations of <a href="/facts/Bell_inequalities/lg1V6kpk">Bell inequalities</a><a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> (in sharp contrast to the bipartite case). The proof uses random unitary matrices. No other proof is available.</li></ul>
<h2 id="information-theory">Information theory</h2>
<ul><li>The proof of <a href="/facts/Claude_Shannon/EremsCws">Shannon</a>'s <a href="/facts/Noisy-channel_coding_theorem/vvr8u4Gg">channel coding theorem</a> uses random coding to show the existence of a code that achieves <a href="/facts/Channel_capacity/55wJYsS6">channel capacity</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Probabilistic_method/AXKXrzrr">Probabilistic method</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://mathoverflow.net/q/9218">Probabilistic Proofs of Analytic Facts</a> at <a href="/facts/MathOverflow/0XqLBHA5">MathOverflow</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Karel de Leeuw, Yitzhak Katznelson and Jean-Pierre Kahane, Sur les coefficients de Fourier des fonctions continues. (French) C. R. Acad. Sci. Paris Sér. A–B 285:16 (1977), A1001–A1003. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Salem, Raphaël (1951). "On singular monotonic functions whose spectrum has a given Hausdorff dimension". Ark. Mat. 1 (4): 353–365. Bibcode:1951ArM.....1..353S. doi:10.1007/bf02591372. <a href="https://doi.org/10.1007%2Fbf02591372" target="_blank">https://doi.org/10.1007%2Fbf02591372</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kaufman, Robert (1981). "On the theorem of Jarník and Besicovitch". Acta Arith. 39 (3): 265–267. doi:10.4064/aa-39-3-265-267. <a href="https://doi.org/10.4064%2Faa-39-3-265-267" target="_blank">https://doi.org/10.4064%2Faa-39-3-265-267</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Blyth, Colin R.; Pathak, Pramod K. (1986), "A note on easy proofs of Stirling's theorem", American Mathematical Monthly, 93 (5): 376–379, doi:10.2307/2323600, JSTOR 2323600. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Gordon, Louis (1994), "A stochastic approach to the gamma function", American Mathematical Monthly, 101 (9): 858–865, doi:10.2307/2975134, JSTOR 2975134. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Revuz, Daniel; Yor, Marc (1994), Continuous martingales and Brownian motion (2nd ed.), Springer (see Exercise (2.17) in Section V.2, page 187). <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">See Fatou's theorem. <a href="/wiki/Fatou%27s_theorem" target="_blank">/wiki/Fatou%27s_theorem</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Durrett, Richard (1984), Brownian motion and martingales in analysis, California: Wadsworth, ISBN 0-534-03065-3. <a href="0-534-03065-3" target="_blank">0-534-03065-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Bass, R.F.; Burdzy, K. (1989), "A probabilistic proof of the boundary Harnack principle", Seminar on Stochastic Processes, Boston: Birkhäuser (published 1990), pp. 1–16, hdl:1773/2249. <a href="/wiki/Richard_F._Bass" target="_blank">/wiki/Richard_F._Bass</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Bass, Richard F. (1995), Probabilistic techniques in analysis, Springer, p. 228. <a href="/wiki/Richard_F._Bass" target="_blank">/wiki/Richard_F._Bass</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Markowsky, Greg T. (2011), "On the expected exit time of planar Brownian motion from simply connected domains", Electronic Communications in Probability, 16: 652–663, arXiv:1108.1188, doi:10.1214/ecp.v16-1653, S2CID 55705658. <a href="http://ecp.ejpecp.org/article/view/1653" target="_blank">http://ecp.ejpecp.org/article/view/1653</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Davis, Burgess (1975), "Picard's theorem and Brownian motion", Transactions of the American Mathematical Society, 213: 353–362, doi:10.2307/1998050, JSTOR 1998050. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Davis, Burgess (1979), "Brownian motion and analytic functions", Annals of Probability, 7 (6): 913–932, doi:10.1214/aop/1176994888. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Wong, T.K.; Yam, S.C.P. (2018), "A probabilistic proof for Fourier inversion formula", Statistics & Probability Letters, 141: 135–142, doi:10.1016/j.spl.2018.05.028, S2CID 125351871. <a href="/w/index.php?title=Wong,_T.K.&action=edit&redlink=1" target="_blank">/w/index.php?title=Wong,_T.K.&action=edit&redlink=1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Abért, Miklós; Weiss, Benjamin (2011). "Bernoulli actions are weakly contained in any free action". arXiv:1103.1063v2 [math.DS]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Revuz, Daniel; Yor, Marc (1994), Continuous martingales and Brownian motion (2nd ed.), Springer (see Exercise (2.17) in Section V.2, page 187). <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Bismut, Jean-Michel (1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem", J. Funct. Anal., 57: 56–99, doi:10.1016/0022-1236(84)90101-0. <a href="/wiki/Jean-Michel_Bismut" target="_blank">/wiki/Jean-Michel_Bismut</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">As long as we have no article on Martin boundary, see Compactification (mathematics)#Other compactification theories. <a href="/wiki/Martin_boundary" target="_blank">/wiki/Martin_boundary</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Bishop, C. (1991), "A characterization of Poissonian domains", Arkiv för Matematik, 29 (1): 1–24, Bibcode:1991ArM....29....1B, doi:10.1007/BF02384328 (see Section 6). <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Tsirelson, Boris (1997), "Triple points: from non-Brownian filtrations to harmonic measures", Geometric and Functional Analysis, 7 (6), Birkhauser: 1096–1142, doi:10.1007/s000390050038, S2CID 121617197. author's site <a href="/wiki/Boris_Tsirelson" target="_blank">/wiki/Boris_Tsirelson</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Tsirelson, Boris (1998), "Within and beyond the reach of Brownian innovation", Proceedings of the international congress of mathematicians, Documenta mathematica, vol. Extra Volume ICM 1998, III, Berlin: der Deutschen Mathematiker-Vereinigung, pp. 311–320, ISSN 1431-0635. <a href="/wiki/Boris_Tsirelson" target="_blank">/wiki/Boris_Tsirelson</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Tolsa, Xavier; Volberg, Alexander (2017). "On Tsirelson's theorem about triple points for harmonic measure". International Mathematics Research Notices. 2018 (12): 3671–3683. arXiv:1608.04022. doi:10.1093/imrn/rnw345. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Horowitz, Charles; Usadi Katz, Karin; Katz, Mikhail G. (2008). "Loewner's torus inequality with isosystolic defect". Journal of Geometric Analysis. 19 (4): 796–808. arXiv:0803.0690. doi:10.1007/s12220-009-9090-y. S2CID 18444111. <a href="/wiki/Mikhail_Katz" target="_blank">/wiki/Mikhail_Katz</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Neel, Robert W. (2008), "A martingale approach to minimal surfaces", Journal of Functional Analysis, 256 (8), Elsevier: 2440–2472, arXiv:0805.0556, doi:10.1016/j.jfa.2008.06.033, S2CID 15228691. Also arXiv:0805.0556. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Fulman, Jason (2001), "A probabilistic proof of the Rogers–Ramanujan identities", Bulletin of the London Mathematical Society, 33 (4): 397–407, arXiv:math/0001078, doi:10.1017/S0024609301008207, S2CID 673691, archived from the original on 2012-07-07. Also arXiv:math.CO/0001078. <a href="https://archive.today/20120707000551/http://blms.oxfordjournals.org/cgi/content/abstract/33/4/397" target="_blank">https://archive.today/20120707000551/http://blms.oxfordjournals.org/cgi/content/abstract/33/4/397</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Arveson, William (2003), Noncommutative dynamics and E-semigroups, New York: Springer, ISBN 0-387-00151-4. <a href="0-387-00151-4" target="_blank">0-387-00151-4</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Tsirelson, Boris (2003), "Non-isomorphic product systems", in Price, Geoffrey (ed.), Advances in quantum dynamics, Contemporary mathematics, vol. 335, American mathematical society, pp. 273–328, ISBN 0-8218-3215-8. Also arXiv:math.FA/0210457. <a href="0-8218-3215-8" target="_blank">0-8218-3215-8</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Tsirelson, Boris (2008), "On automorphisms of type II Arveson systems (probabilistic approach)", New York Journal of Mathematics, 14: 539–576. <a href="/wiki/Boris_Tsirelson" target="_blank">/wiki/Boris_Tsirelson</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Bhat, B.V.Rajarama; Srinivasan, Raman (2005), "On product systems arising from sum systems", Infinite Dimensional Analysis, Quantum Probability and Related Topics, 8 (1): 1–31, arXiv:math/0405276, doi:10.1142/S0219025705001834, S2CID 15106610. Also arXiv:math.OA/0405276. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Izumi, Masaki; Srinivasan, Raman (2008), "Generalized CCR flows", Communications in Mathematical Physics, 281 (2): 529–571, arXiv:0705.3280, Bibcode:2008CMaPh.281..529I, doi:10.1007/s00220-008-0447-z, S2CID 12815055. Also arXiv:0705.3280. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Perez-Garcia, D.; Wolf, M.M.; C., Palazuelos; Villanueva, I.; Junge, M. (2008), "Unbounded violation of tripartite Bell inequalities", Communications in Mathematical Physics, 279 (2): 455–486, arXiv:quant-ph/0702189, Bibcode:2008CMaPh.279..455P, doi:10.1007/s00220-008-0418-4, S2CID 29110154 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
</ol>

List of probabilistic proofs of non-probabilistic theorems open-in-new

List of probabilistic proofs of non-probabilistic theorems