Rademacher distribution

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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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

References[edit]

  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