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.
Mathematical formulation The probability mass function of this distribution is
f ( k ) = { 1 / 2 if k = − 1 , 1 / 2 if k = + 1 , 0 otherwise. {\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\text{if }}k=-1,\\1/2&{\text{if }}k=+1,\\0&{\text{otherwise.}}\end{matrix}}\right.}
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
Pr ( ∑ i x i a i > t ‖ a ‖ 2 ) ≤ e − t 2 / 2 {\displaystyle \Pr \left(\sum _{i}x_{i}a_{i}>t\|a\|_{2}\right)\leq e^{-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. 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
Pr ( ‖ Y ‖ > s t ) ≤ [ 1 c Pr ( ‖ Y ‖ > t ) ] c s 2 {\displaystyle \Pr \left(\|Y\|>st\right)\leq \left[{\frac {1}{c}}\Pr(\|Y\|>t)\right]^{cs^{2}}}
for some constant c. Let p be a positive real number. Then the Khintchine inequality says that
c 1 [ ∑ | a i | 2 ] 1 2 ≤ ( E [ | ∑ a i x i | p ] ) 1 p ≤ c 2 [ ∑ | a i | 2 ] 1 2 {\displaystyle c_{1}\left[\sum \left|a_{i}\right|^{2}\right]^{\frac {1}{2}}\leq \left(E\left[\left|\sum a_{i}x_{i}\right|^{p}\right]\right)^{\frac {1}{p}}\leq c_{2}\left[\sum \left|a_{i}\right|^{2}\right]^{\frac {1}{2}}}
where c1 and c2 are constants dependent only on p. For p ≥ 1, c 2 ≤ c 1 p . {\displaystyle c_{2}\leq c_{1}{\sqrt {p}}.}
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