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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

$f(k)=\left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.$

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

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

van Zuijlen's bound

van Zuijlen has proved the following result.

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

$\Pr {\Bigl (}{\Bigl |}{\frac {\sum _{i=1}^{n}X_{i}}{\sqrt {n}}}{\Bigr |}\leq 1)\geq 0.5.$

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

$\Pr(\sum _{i}X_{i}a_{i}>t||a_{i}||_{2})\leq e^{-{\frac {t^{2}}{2}}}$

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

$\Pr(\sum _{i}X_{i}a_{i}>t||a_{i}||_{1})=0$

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

$Pr(||Y||>st)\leq [{\frac {1}{c}}Pr(||Y||>t)]^{cs^{2}}$

for some constant c.

Let p be a positive real number. Then

$c_{1}[\sum {|a_{i}|^{2}}]^{\frac {1}{2}}\leq (E[|\sum {a_{i}X_{i}}|^{p}])^{\frac {1}{p}}\leq c_{2}[\sum {|a_{i}|^{2}}]^{\frac {1}{2}}$

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1

$c_{2}\leq c_{1}{\sqrt {p}}$

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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