Rademacher distribution
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In
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.
Mathematical formulation
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
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
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[3]
for some constant c.
Let p be a positive real number. Then the Khintchine inequality says that[4]
where c1 and c2 are constants dependent only on p.
For p ≥ 1,
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[5][6] culminated into a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.[7]
Conjecture. Let , where and the 's are independent Rademacher variables. Then
For example, when , one gets the following bound, first shown by Van Zuijlen.[8]
The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31).
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,[9] 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.
- numerical optimization.
Rademacher random variables are used in the Symmetrization Inequality.
Related distributions
- Bernoulli distribution: If X has a Rademacher distribution, then has a Bernoulli(1/2) distribution.
- Laplace distribution: If X has a Rademacher distribution and Y ~ Exp(λ) is independent from X, then XY ~ Laplace(0, 1/λ).
References
- ISBN 978-1-4612-6682-2.
- .
- S2CID 15159626.
- S2CID 119840766.
- S2CID 20281665.
- arXiv:1704.00350 [math.CO].
- arXiv:2006.16834 [math.CO].
- )
- S2CID 5827717.