Rademacher distribution

Bounds on sums of independent Rademacher variables

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.

Concentration inequalities

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

${\displaystyle Pr\left(\sum _i{x}_i{a}_i>t\Vert a{\Vert }_2\right)\le e^{-t^2/2}}$

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

${\displaystyle Pr\left(\Vert Y\Vert >st\right)\le {\left[\frac{1}{c}Pr(\Vert Y\Vert >t)\right]}^{c{s}^2}}$

for some constant c.

Let p be a positive real number. Then the Khintchine inequality says that

${\displaystyle c_1{\left[\sum {\left|a_i\right|}^2\right]}^{\frac{1}{2}}\le {\left(E\left[{\left|\sum a_i{x}_i\right|}^p\right]\right)}^{\frac{1}{p}}\le c_2{\left[\sum {\left|a_i\right|}^2\right]}^{\frac{1}{2}}}$

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1, ${\displaystyle c_2\le c_1\sqrt{p}.}$

Tomaszewski’s conjecture

In 1986, Bogusław Tomaszewski proposed a question about the distribution of the sum of independent Rademacher variables. A series of works on this question culminated in a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.

Conjecture. Let ${\displaystyle X=\sum _{i=1}^n{a}_i{X}_i}$, where ${\displaystyle a_1^2+\cdots +a_n^2=1}$ and the ${\displaystyle X_i}$'s are independent Rademacher variables. Then

${\displaystyle Pr[|X|\le 1]\ge 1/2.}$

For example, when ${\displaystyle a_1=a_2=\cdots =a_n=1/\sqrt{n}}$, one gets the following bound, first shown by Van Zuijlen.

${\displaystyle Pr\left(\left|\frac{\sum _{i=1}^n{X}_i}{\sqrt{n}}\right|\le 1\right)\ge \mathrm{0.5.}}$

The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31).

Applications

The Rademacher distribution has been used in bootstrapping. See Chapter 17 of Testing Statistical Hypotheses for example. The distribution is particularly useful in high-dimensional statistics.

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:

• The Hutchinson trace estimator, which can be used to efficiently approximate the trace of a matrix of which the elements are not directly accessible, but rather implicitly defined via matrix-vector products.
• SPSA, a computationally cheap, derivative-free, stochastic gradient approximation, useful for numerical optimization.

Rademacher random variables are used in the Symmetrization Inequality.

Related distributions

• Bernoulli distribution: If X has a Rademacher distribution, then $(X+1)/2$ has a Bernoulli(1/2) distribution.
• Laplace distribution: If X has a Rademacher distribution and Y ~ Exp(λ) is independent from X, then XY ~ Laplace(0, 1/λ).