rademacher distribution

Author

darshan gowda v

Abstract:-

The Rademacher distribution is a recoding of the Bernoulli distribution with two possible values {-1, 1}. It’s second moment (the variance) equals 1; all other moments equal 0 . It is named after German-American mathematician Hans Rademacher and denoted Rad½.

Like the Bernoulli, a random variable has a 50% chance of a success and 50% chance of failure.

Bernoulli: 0 (failure) and 1 (success),

Rademacher: -1 (failure) and 1 (success).

The distribution is used for formulating statistical proofs, random sampling , and bootstrapping , where weights dg = {−1, 1} are called Rademacher weights

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.also used to efficiently approximate the trace of a matrix

Introduction:-

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.

A series (that is, a sum) of Rademacher distributed variables can be regarded as a simple symmetrical random walk where the step size is 1.

Bernoulli distribution: If X has a Rademacher distribution, then (X+1)/2 has a Bernoulli(1/2) distribution.

Formulas of the distribution:-

The probability mass function of this distribution is

In terms of the Dirac delta function, as

Bounds on sums of independent Rademacher variables:

There are various results in probability theory around analyzing the sum of i.i.d. Rademacher variables, including concentration inequalities such as Bernstein inequalities as well as anti-concentration inequalities like Tomaszewski’s conjecture.

Concentration inequalities:

Let {xi} be a set of random variables with a Rademacher distribution. Let {ai} be a sequence of real numbers. Then

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.

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

for some constant c.

Let p be a positive real number. Then the Khintchine inequality says that

where c1 and c2 are constants dependent only on p.

Tomaszewski’s conjecture:

In 1986, Bogusław Tomaszewski proposed a question about the distribution of the sum of independent Rademacher variables.

A series of works on this question

culminated into a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.

The bound is sharp and better than that which can be derived from the normal distribution

real life application of distribution:-

(To determine )

1)Number of Side Effects from Medications

2)Shopping Returns per Week

3)Number of Spam Emails per Day

4)Number of River Overflows

Problems and solutions related to the distribution:-

1)interpret the sharpness of the bound and compare that with normal distribution for the given

data ,where (a1)^2+(a2) ^2+ ………+ (a4) ^2=1,and X=0.5,1,0,0.5.

solution:

given n=4

X=0.5,1,0,0.5

wkt from Tomaszewski’s conjecture:

Pr(((0.5+1+.5)/2)<=1)

Pr((2/2)<=1)

Pr(1)

which is >0.5

hence from Tomaszewski’s conjecture:

The bound is sharp and better than that which can be derived from the normal distribution

Conclusion:-

Bernoulli distribution: If X has a Rademacher distribution, then (X+1)/2 has a Bernoulli(1/2) distribution.

2)The Rademacher distribution can be used to show that normally distributed and uncorrelated does not imply independent.

Reference:-

  1. Hitczenko, P.; Kwapień, S. (1994). “On the Rademacher series”. Probability in Banach Spaces. Progress in probability. Vol. 35. pp. 31–36. doi:10.1007/978-1-4612-0253-0_2. ISBN 978-1-4612-6682-2.

  2. Dilworth, S. J.; Montgomery-Smith, S. J. (1993). “The distribution of vector-valued Radmacher series”. Ann Probab. 21 (4): 2046–2052. arXiv:math/9206201. doi:10.1214/aop/1176989010. JSTOR 2244710. S2CID 15159626.