Rademacher distribution

Rademacher
Support
pmf
CDF
Mean
Median
Mode	N/A
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

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 +1 and a 50% chance of being -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.

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 \left(\left|{\frac {\sum _{i=1}^{n}X_{i}}{\sqrt {n}}}\right|\leq 1\right)\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 \left(\sum _{i}x_{i}a_{i}>t||a||_{2}\right)\leq 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 \left(||Y||>st\right)\leq \left[{\frac {1}{c}}\Pr(||Y||>t)\right]^{cs^{2}}

for some constant c.

Let p be a positive real number. Then the Khintchine inequality says that^[5]

c_{1}\left[\sum {\left|a_{i}\right|^{2}}\right]^{\frac {1}{2}}\leq \left(E\left[\left|\sum {a_{i}x_{i}}\right|^{p}\right]\right)^{\frac {1}{p}}\leq 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,

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

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". Probability in Banach Spaces. Progress in probability. 35. pp. 31–36. doi:10.1007/978-1-4612-0253-0_2.
↑ van Zuijlen, Martien C. A. (2011). "On a conjecture concerning the sum of independent Rademacher random variables". arXiv:1112.4988. Bibcode:2011arXiv1112.4988V.
↑ Montgomery-Smith, S. J. (1990). "The distribution of Rademacher sums". Proc Amer Math Soc. 109: 517–522. doi:10.1090/S0002-9939-1990-1013975-0.
↑ Dilworth, S. J.; Montgomery-Smith, S. J. (1993). "The distribution of vector-valued Radmacher series". Ann Probab. 21 (4): 2046–2052. JSTOR 2244710.
↑ Khintchine, A. (1923). "Über dyadische Brüche". Math. Z. 18 (1): 109–116. doi:10.1007/BF01192399.
↑ Avron, H.; Toledo, S. (2011). "Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix". Journal of the ACM. 58 (2): 8. doi:10.1145/1944345.1944349.

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[Hitczenko1994-1] Hitczenko, P.; Kwapień, S. (1994). "On the Rademacher series". Probability in Banach Spaces. Progress in probability. 35. pp. 31–36. doi:10.1007/978-1-4612-0253-0_2.

[vanZuijlen2011-2] van Zuijlen, Martien C. A. (2011). "On a conjecture concerning the sum of independent Rademacher random variables". arXiv:1112.4988. Bibcode:2011arXiv1112.4988V.

[MontgomerySmith1990-3] Montgomery-Smith, S. J. (1990). "The distribution of Rademacher sums". Proc Amer Math Soc. 109: 517–522. doi:10.1090/S0002-9939-1990-1013975-0.

[Dilworth1993-4] Dilworth, S. J.; Montgomery-Smith, S. J. (1993). "The distribution of vector-valued Radmacher series". Ann Probab. 21 (4): 2046–2052. JSTOR 2244710.

[Khintchine1923-5] Khintchine, A. (1923). "Über dyadische Brüche". Math. Z. 18 (1): 109–116. doi:10.1007/BF01192399.

[6] Avron, H.; Toledo, S. (2011). "Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix". Journal of the ACM. 58 (2): 8. doi:10.1145/1944345.1944349.

Probability distributions
List
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 Lomax 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 Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's 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 Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace 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

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	$\cos(t)\,$