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Probability theory routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs. ==Analysis== * Normal numbers exist. Moreover, computable normal numbers exist. 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 strong law of large numbers); (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. * Dvoretzky's theorem which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies. * The diameter of the Banach–Mazur compactum was calculated using a probabilistic construction. No deterministic construction is known. * The original proof that the Hausdorff–Young inequality cannot be extended to is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.〔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.〕 * The first construction of a Salem set was probabilistic. Only in 1981 did Kaufman give a deterministic construction. * Every continuous function on an compact interval can be uniformly approximated by polynomials, which is the Weierstrass approximation theorem. A probabilistic proof uses the weak law of large numbers. Non-probabilistic proofs were available earlier. * Existence of a nowhere differentiable continuous function follows easily from properties of Wiener process. A non-probabilistic proof was available earlier. * Stirling's formula was first discovered by Abraham de Moivre in his `The Doctrine of Chances' (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.〔.〕〔.〕 * The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem. A probabilistic proof via two-dimensional Brownian motion is well-known.〔 (see Exercise (2.17) in Section V.2, page 187).〕 Non-probabilistic proofs were available earlier. * Non-tangential boundary values〔See Fatou's theorem.〕 of an analytic or harmonic function exist at almost all boundary points of non-tangential boundedness. This result (Privalov's theorem), and several results of this kind, are deduced from martingale convergence.〔.〕 Non-probabilistic proofs were available earlier. * The boundary Harnack principle is proved using Brownian motion〔.〕 (see also〔.〕). Non-probabilistic proofs were available earlier. * Euler's Basel sum, 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.〔.〕 * The Picard Theorem can be proved using the winding properties of planar Brownian motion.〔.〕〔.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of probabilistic proofs of non-probabilistic theorems」の詳細全文を読む スポンサード リンク
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