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Rademacher distribution

 

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.

Contents

  • Mathematical formulation 1
  • van Zuijlen's bound 2
  • Bounds on sums 3
  • Applications 4
  • Related distributions 5
  • References 6

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 Xi 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 { Xi } be a set of random variables with a Rademacher distribution. Let { ai } 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 { ai }, t > 0 is a real number and Pr(Z) is the probability of event Z.[3]

Let Y = Σ Xiai 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 }

Another bound on the sums is known as the Bernstein inequalities.

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.

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

  1. ^ Hitczenko P, Kwapień S (1994) On the Rademacher series. Progress in probability 35: 31-36
  2. ^ van Zuijlen Martien CA (2011) On a conjecture concerning the sum of independent Rademacher random variables. http://arxiv.org/abs/1112.4988
  3. ^ MontgomerySmith SJ (1990) The distribution of Rademacher sums. Proc Amer Math Soc 109: 517522
  4. ^ Dilworth SJ, Montgomery-Smith SJ (1993) The distribution of vector-valued Radmacher series. Ann Probab 21 (4) 2046-2052
  5. ^ Khintchine A (1923) Über dyadische Brüche. Math Zeitschr 18: 109–116
  6. ^ 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.
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