Probability Mass of Rademacher Sums Beyond One Standard Deviation

Statistics at square one. III--Standard deviation.

Rademacher sums, Moonshine and Gravity

In 1939 Rademacher demonstrated how to express Klein’s modular invariant as a sum over elements of the modular group. In this article we generalize Rademacher’s approach so as to construct bases for the spaces of automorphic integrals of arbitrary even integer weight, for an arbitrary group commensurable with the modular group. Our methods provide explicit expressions for the Fourier expansions...

Lévy’s inequality, Rademacher sums, and Kahane’s inequality

Let (Ω,A , P ) be a probability space. A random variable is a Borel measurable function Ω→ R. For a random variable X, we denote by X∗P the pushforward measure of P by X. X∗P is a Borel probability measure on R, called the distribution of X. A random variable X is called symmetric when the distribution of X is equal to the distribution of −X. Because the collection {(−∞, a] : a ∈ R} generates t...

A Bound on the Deviation Probability for Sums of Non-negative Random Variables

A simple bound is presented for the probability that the sum of nonnegative independent random variables is exceeded by its expectation by more than a positive number t. If the variables have the same expectation the bound is slightly weaker than the Bennett and Bernstein inequalities, otherwise it can be significantly stronger. The inequality extends to one-sidedly bounded martingale differenc...

On the Tail Estimation of the Norm of Rademacher Sums

The main aim of this paper is to prove a bilateral inequality for P [‖ n ∑ 1 akrk‖ > t ] , where t > 0, (ak) are elements of a normed space, while (rk) are Rademacher functions. Then this inequality is applied for estimation of E‖ n ∑ 1 akrk‖. As another corollary we give a maximal inequality for exchangeable random variables that recently has been published in [4]. 2000 Mathematics Subject Cla...