Rademacher distribution

The probability mass function of this distribution is

:

1/2 & \text{if }k=+1, \\

0 & \text{otherwise.}\end{matrix}\right.

In terms of the Dirac delta function, as

:$f(\; k\; )\; =\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; (\; \backslash delta\; (k\; -\; 1)\; +\; \backslash delta\; (k\; +\; 1)).$

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.

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

:$\backslash Pr\backslash left(\; \backslash sum\_i\; x\_i\; a\_i\; >\; t\; \backslash |\; a\; \backslash |\_2\; \backslash right)\; \backslash le\; e^\{\; -t^2/2\; \}$

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.{{cite journal |last=Montgomery-Smith |first=S. J. |year=1990 |title=The distribution of Rademacher sums |journal=Proc Amer Math Soc |volume=109 |issue= 2|pages=517–522 |doi=10.1090/S0002-9939-1990-1013975-0 |doi-access=free }}

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{{cite journal |last1=Dilworth |first1=S. J. |last2=Montgomery-Smith |first2=S. J. |year=1993 |title=The distribution of vector-valued Radmacher series |journal=Ann Probab |volume=21 |issue=4 |pages=2046–2052 |jstor=2244710 |doi=10.1214/aop/1176989010 |arxiv=math/9206201 |s2cid=15159626 }}

:$\backslash Pr\backslash left(\; \backslash |\; Y\; \backslash |\; >\; st\; \backslash right)\; \backslash le\; \backslash left[\; \backslash frac\{\; 1\; \}\{\; c\; \}\; \backslash Pr(\; \backslash |\; Y\; \backslash |\; >\; t\; )\; \backslash right]^\{\; cs^2\; \}$

for some constant c.

Let p be a positive real number. Then the Khintchine inequality says that{{cite journal |last=Khintchine |first=A. |year=1923 |title=Über dyadische Brüche |journal=Math. Z. |volume=18 |issue=1 |pages=109–116 |doi=10.1007/BF01192399 |s2cid=119840766 }}

:$c\_1\; \backslash left[\; \backslash sum\; \backslash left|\; a\_i\; \backslash right|^2\; \backslash right]^\backslash frac\{\; 1\; \}\{\; 2\; \}\; \backslash le\; \backslash left(\; E\backslash left[\; \backslash left|\; \backslash sum\; a\_i\; x\_i\; \backslash right|^p\; \backslash right]\; \backslash right)^\{\; \backslash frac\{\; 1\; \}\{\; p\; \}\; \}\; \backslash le\; c\_2\; \backslash left[\; \backslash sum\; \backslash left|\; a\_i\; \backslash right|^2\; \backslash right]^\backslash frac\{\; 1\; \}\{\; 2\; \}$

where c₁ and c₂ are constants dependent only on p.

For p ≥ 1, $c\_2\; \backslash le\; c\_1\; \backslash sqrt\{\; p\; \}.$

In 1986, Bogusław Tomaszewski proposed a question about the distribution of the sum of independent Rademacher variables. A series of works on this question{{Cite journal|last1=Holzman|first1=Ron|last2=Kleitman|first2=Daniel J.|date=1992-09-01|title=On the product of sign vectors and unit vectors|url=https://doi.org/10.1007/BF01285819|journal=Combinatorica|language=en|volume=12|issue=3|pages=303–316|doi=10.1007/BF01285819|s2cid=20281665|issn=1439-6912}}{{cite arXiv|last1=Boppana|first1=Ravi B.|last2=Holzman|first2=Ron|date=2017-08-31|title=Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier|class=math.CO|eprint=1704.00350}} culminated into a proof in 2020 by Nathan Keller and Ohad Klein of the following conjecture.{{cite arXiv|last1=Keller|first1=Nathan|last2=Klein|first2=Ohad|date=2021-08-03|title=Proof of Tomaszewski's Conjecture on Randomly Signed Sums|class=math.CO|eprint=2006.16834}}

Conjecture. Let $X\; =\; \backslash sum\_\{i=1\}^n\; a\_i\; X\_i$, where $a\_1^2\; +\; \backslash cdots\; +\; a\_n^2\; =\; 1$ and the $X\_i$'s are independent Rademacher variables. Then

:$\backslash Pr[|X|\; \backslash leq\; 1]\; \backslash geq\; 1/2.$

For example, when $a\_1\; =\; a\_2\; =\; \backslash cdots\; =\; a\_n\; =\; 1/\backslash sqrt\{n\}$, one gets the following bound, first shown by Van Zuijlen.{{cite arXiv |last=van Zuijlen |first=Martien C. A. |year=2011 |title=On a conjecture concerning the sum of independent Rademacher random variables |class=math.PR |eprint=1112.4988 }}

: $\backslash Pr\; \backslash left(\; \backslash left|\; \backslash frac\{\; \backslash sum\_\{i\; =\; 1\}^n\; X\_i\; \}\; \backslash sqrt\; n\; \backslash right|\; \backslash le\; 1\; \backslash right)\; \backslash ge\; 0.5.$

The bound is sharp and better than that which can be derived from the normal distribution (approximately Pr > 0.31).

The Rademacher distribution has been used in bootstrapping. See Chapter 17 of Testing Statistical Hypotheses for example.{{cite book |last1=Lehmann |first1=Erich L. |last2=Romano |first2=Joseph P. |title=Testing statistical hypotheses |date=2022 |publisher=Springer |location=Cham |isbn=978-3-030-70577-0 |pages=835–842 |edition=Forth |url=https://link.springer.com/book/10.1007/978-3-030-70578-7}}

The distribution is particularly useful in high-dimensional statistics.{{cite journal |last1=Chernozhuokov |first1=Victor |last2=Chetverikov |first2=Denis |last3=Kato |first3=Kengo |last4=Koike |first4=Yuta |title=Improved central limit theorem and bootstrap approximations in high dimensions |journal=Annals of Statistics |date=October 2022 |volume=50 |issue=5 |pages=2562 -- 2586 |doi=10.1214/22-AOS2193|arxiv=1912.10529 }}

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,{{cite journal |last1=Avron |first1=H. |last2=Toledo |first2=S. |title=Randomized algorithms for estimating the trace of an implicit symmetric positive semidefinite matrix |journal=Journal of the ACM |volume=58 |issue=2 |pages=8 |year=2011 |doi=10.1145/1944345.1944349 |citeseerx=10.1.1.380.9436 |s2cid=5827717 }} 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.

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

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Category:Discrete distributions

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