Rademacher distribution

A Rademacher random variable takes the values $1$ and $-1$, each with probability $1/2$.

On $[0,1)$ with Lebesgue measure, the functions $r_n(t)=(-1{)}^{\lfloor 2^{n}t\rfloor }$, $n\ge 1$, are independent symmetric Rademacher random variables. Every prescribed pattern for the first $n$ functions occupies a dyadic interval of measure $2^{-n}$. Their closed span in $L^1$ is isomorphic to the l2 sequence space by the Khintchine inequality.

A Rademacher sum is a linear combination of independent Rademacher random variables, with scalar or normed-space-valued coefficients. Its moments are controlled by the Khintchine inequality in the scalar case and the Kahane-Khintchine inequality in a general normed vector space.

For independent Rademacher random variables, this series converges absolutely on every sign sequence and has the uniform distribution on $[-1,1]$. Its first $n$ terms are uniform on the $2^n$ midpoints of equal subintervals of $[-1,1]$, giving weak convergence of probability measures by Riemann sums. The remaining tail has absolute value at most $2^{-n}$, and its mean square is $4^{-n}/3$ by independence. Thus convergence is also almost surely and in $L^2$.

On the Boolean hypercube $\{-1,1{\}}^d$ with uniform measure, each coordinate is a Rademacher random variable. Products of these sign coordinate functions are Walsh characters. This use of Bernoulli function is distinct from the fluid-mechanical Bernoulli function.

For independent Rademacher random variables and $\Vert u_i{\Vert }_2\le 1$, this bound holds for $\alpha >0$. The moment-generating function satisfies ${\prod }_j\cosh (t{u}_{ij})\le e^{t^2/2}$, with strict inequality for a nonzero vector and $t>0$. Applying the exponential Markov bound to both tails and then the union bound proves a strict failure bound; a zero vector has failure probability zero. At $\alpha =\sqrt{2\log (2m)}$, the success probability is strictly positive, ensuring a sign vector with all linear forms below that threshold.