Probabilistic Proofs of Non-probabilistic Theorems - Analysis

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.

• The first construction of a Salem set was probabilistic. Only in 1981 did Kaufman give a deterministic construction.

• Every continuous function on an 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. Non-probabilistic proofs were available earlier.

• Non-tangential boundary values 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.