Rademacher distribution

Rademacher
Support	$k \in \{-1,1\}\,$
pmf	$f(k) = \left\{\begin{matrix} 1/2 & \mbox {if }k=-1, \\ 1/2 & \mbox {if }k=+1, \\ 0 & \mbox {otherwise.}\end{matrix}\right.$
CDF	$F(k) = \begin{cases} 0, & k < -1 \\ 1/2, & -1 \leq k < 1 \\ 1, & k \geq 1 \end{cases}$
Mean	$0\,$
Median	$0\,$
Mode	N/A
Variance	$1\,$
Skewness	$0\,$
Ex. kurtosis	$-2\,$
Entropy	$\ln(2)\,$
MGF	$\cosh(t)\,$
CF	$\cos(t)\,$

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.^[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.^[2]

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| \le 1 \Bigr) \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

\Pr( \sum_i X_i a_i > t || a ||_2 ) \le e^{ - \frac{ 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.^[3]

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^[4]

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

for some constant c.

Let p be a positive real number. Then^[5]

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

$c_2 \le c_1 \sqrt{ p }$

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:

The Hutchinson trace estimator,^[6] 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 $\frac{X+1}{2}$ has a Bernoulli(1/2) distribution.
Laplace distribution: If X has a Rademacher distribution and Y ~ Exp(λ), then XY ~ Laplace(0, 1/λ).

References

↑ Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36
↑ van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. http://arxiv.org/abs/1112.4988
↑ MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522
↑ Dilworth SJ, Montgomery-Smith SJ (1993) The distribution of vector-valued Radmacher series. Ann Probab 21 (4) 2046-2052
↑ Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18: 109–116
↑ Avron, H. and Toledo, S. Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix. Journal of the ACM, 58(2):8, 2011.

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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