Rademacher distribution


In probability theory and statistics, the Rademacher distribution 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 of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1.

Mathematical formulation

The probability mass function of this distribution is
In terms of the Dirac delta function, as

Van Zuijlen's bound

Van Zuijlen has proved the following result.
Let Xi be a set of independent Rademacher distributed random variables. Then
The bound is sharp and better than that which can be derived from the normal distribution.

Bounds on sums

Let be a set of random variables with a Rademacher distribution. Let be a sequence of real numbers. Then


where ||a||2 is the Euclidean norm of the sequence, t > 0 is a real number and Pr is the probability of event Z.
Let Y = Σ xiai 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.
Let p be a positive real number. Then the Khintchine inequality says that
where c1 and c2 are constants dependent only on p.
For p ≥ 1,
See also: Concentration inequality - a summary of tail-bounds on random variables.

Applications

The Rademacher distribution has been used in bootstrapping.
The Rademacher distribution can be used to show that normally distributed and uncorrelated does not imply independent.
Random vectors with components sampled independently from the Rademacher distribution are useful for various stochastic approximations, for example:
Rademacher random variables are used in the Symmetrization Inequality.

Related distributions