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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 either +1 or -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

$f(k)=\{\begin{array}{cc}1/2& \text{if }k=-1,\\ 1/2& \text{if }k=+1,\\ 0& \text{otherwise.}\end{array}$

It can be also written as a probability density function, in terms of the Dirac delta function, as

$f(k)=\frac{1}{2}(\delta (k-1)+\delta (k+1)).$

van Zuijlen's bound

van Zuijlen has proved the following result.

Let X_i be a set of independent Rademacher distributed random variables. Then

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

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

Bounds on sums

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

$\mathrm{Pr}(\sum _i{X}_i{a}_i>t|\left|a_i\right|{|}_2)\le e^{-\frac{t^2}{2}}$

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

Also if ||a_i||₁ is finite then

$\mathrm{Pr}(\sum _i{X}_i{a}_i>t|\left|a_i\right|{|}_1)=0$

where || a_i ||₁ is the 1-norm of the sequence { a_i }.

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

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

for some constant c.

Let p be a positive real number. Then

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

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1

$c_2\le c_1\sqrt{p}$

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.

Related distributions

• Bernoulli distribution: If X has a Rademacher distribution then $\frac{X+1}{2}$ has a Bernoulli(1/2) distribution.

References

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