Generalizations of Cauchy-schwarz in Probability Theory

We explore two generalizations of the Cauchy-Schwarz Bessel’s inequality and the Selberg inequality and their application to probability theory. We then give a tautological proof of the De Caen-Selberg Inequality and a proof of the second Borel-Cantelli Lemma with negative dependence. We finish with a suggestion of how linear operator theory can help us understand the tightness of many Selberg-type bounds. 1. Improving on the Cauchy-Schwarz Inequality While studying any well structured space knowing how to control the size of ‘correlation’ between two elements is an essential tool. If we are confined to a vector space X with an inner product, where we measure this correlation by the inner product (〈·, ·〉), the first such control is given by the triangle inequality. A better and hugely more successful tool is the Cauchy-(Buniakowsky-)Schwarz inequality. The trick to using the Cauchy-Schwarz inequality effectively is to choose the right pair of vectors on which to apply it. A clue as to which pair one should choose is to keep in mind that the equality holds when one vector is a constant multiple of the other. To emphasize this point a bit more, consider the case in which x ∈ R and {y1, y2, . . . , yk} ⊂ R is a collection of orthonormal vectors. Then, if we are interested in controlling the size of ∑k j=1 |〈x, yi〉| 2 a direct use of the Cauchy-Schwarz inequality would give us a bound of k||x||. But, if we write x = ∑k j=1 〈x, yj〉 yj+z, where z is orthogonal to each of the yj ’s then we get the inequality, (1) k ∑ j=1 |〈x, yj〉| ≤ ‖x‖ . We immediately see a gain of a factor of k. This gain comes from the fact that a fixed vector x cannot be simultaneously close in direction to multiple yi. But, x can be well approximated by a linear combination of the yi’s. Considering this, we use the Cauchy-Schwarz inequality on the 〈 x, ∑k j=1 〈x, yj〉 yj 〉 to get the better bound in (1) rather than using Cauchy-Schwarz on each of the terms individually. The inequality in Equation 1 is called Bessel’s inequality and is the primary tool in Hilbert space theory to establish the existence of a basis. R. Hasegawa, B. Karmakar: The Wharton School, Department of Statistics, Huntsman Hall 4th Floor, University of Pennsylvania, Philadelphia, PA 19104. Email address: [email protected], [email protected]. 1