Rademacher distribution

In probability theory and statistics, the Rademacher distribution (which is named after Hans Rademacher) is a discrete probability distribution where a random variate X has a 50% chance of being +1 and a 50% chance of being −1.

A series (that is, a sum) of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1.

The probability mass function of this distribution is

There are various results in probability theory around analyzing the sum of i.i.d. Rademacher variables, including concentration inequalities such as Bernstein inequalities as well as anti-concentration inequalities like Tomaszewski's conjecture.

Let {x_i} be a set of random variables with a Rademacher distribution. Let {a_i} be a sequence of real numbers. Then

where ||a||₂ is the Euclidean norm of the sequence {a_i}, t > 0 is a real number and Pr(Z) is the probability of event Z.

Let Y = Σ x_ia_i and let Y be an almost surely convergent series in a Banach space. The for t > 0 and s ≥ 1 we have

for some constant c.