The Khintchine Inequality

For a Rademacher sequence {X=(X_1,X_2,\ldots)} and square summable sequence of real numbers {a=(a_1,a_2,\ldots)}, the Khintchine inequality provides upper and lower bounds for the moments of the random variable,

\displaystyle  a\cdot X=a_1X_1+a_2X_2+\cdots.

We use {\ell^2} for the space of square summable real sequences and

\displaystyle  \lVert a\rVert_2=\left(a_1^2+a_2^2+\cdots\right)^{1/2}

for the associated Banach norm.

Theorem 1 (Khintchine) For each {0 < p < \infty}, there exists positive constants {c_p,C_p} such that,

\displaystyle  c_p\lVert a\rVert_2^p\le{\mathbb E}\left[\lvert a\cdot X\rvert^p\right]\le C_p\lVert a\rVert_2^p, (1)

for all {a\in\ell^2}.

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Rademacher Series

The Rademacher distribution is probably the simplest nontrivial probability distribution that you can imagine. This is a discrete distribution taking only the two possible values {\{1,-1\}}, each occurring with equal probability. A random variable X has the Rademacher distribution if

\displaystyle  {\mathbb P}(X=1)={\mathbb P}(X=-1)=1/2.

A Randemacher sequence is an IID sequence of Rademacher random variables,

\displaystyle  X = (X_1,X_2,X_3\ldots).

Recall that the partial sums {S_N=\sum_{n=1}^NX_n} of a Rademacher sequence is a simple random walk. Generalizing a bit, we can consider scaling by a sequence of real weights {a_1,a_2,\ldots}, so that {S_N=\sum_{n=1}^Na_nX_n}. I will concentrate on infinite sums, as N goes to infinity, which will clearly include the finite Rademacher sums as the subset with only finitely many nonzero weights.

Rademacher series serve as simple prototypes of more general IID series, but also have applications in various areas. Results include concentration and anti-concentration inequalities, and the Khintchine inequality, which imply various properties of {L^p} spaces and of linear maps between them. For example, in my notes constructing the stochastic integral starting from a minimal set of assumptions, the {L^0} version of the Khintchine inequality was required. Rademacher series are also interesting in their own right, and a source of some very simple statements which are nevertheless quite difficult to prove, some of which are still open problems. See, for example, Some explorations on two conjectures about Rademacher sequences by Hu, Lan and Sun. As I would like to look at some of these problems in the blog, I include this post to outline the basic constructions. One intriguing aspect of Rademacher series, is the way that they mix discrete distributions with combinatorial aspects, and continuous distributions. On the one hand, by the central limit theorem, Rademacher series can often be approximated well by a Gaussian distribution but, on the other hand, they depend on the discrete set of signs of the individual variables in the sum. Continue reading “Rademacher Series”