
As anyone who has worked in probability knows, a random variable can sometimes behave in rather “unreasonable” ways. It may be never close to its expectation. It might exceed its expectation almost always, or almost never. It might have finite $1$st, $2$nd, and $3$rd moments, but an infinite $4$th moment. All of this poor behaviour can cause a lot of trouble — wouldn’t it be nice to have a class of “reasonable” random variables?
Continue reading §9.1: Lowdegree polynomials are reasonable
In 1970, Bonami proved the following central result:
The Hypercontractivity Theorem Let $f : \{1,1\}^n \to {\mathbb R}$ and let ${1 \leq p \leq q \leq \infty}$. Then $\\mathrm{T}_\rho f\_q \leq \f\_p$ for $0 \leq \rho \leq \sqrt{\tfrac{p1}{q1}}$.
Continue reading Chapter 9: Basics of hypercontractivity
The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]).
Continue reading Chapter 8 notes
A decision tree $T$ for $f : \{1,1\}^n \to \{1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they are equal, output their value; otherwise, query and output $x_3$.” For a worstcase input (one where $x_1 \neq x_2$) this algorithm has a cost of $3$, meaning it makes $3$ queries. The cost of the worstcase input is the depth of the decision tree.
Continue reading §8.6: Highlight: Randomized decision tree complexity
The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $\Omega = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $\Omega \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, \mathsf{Green}, \mathsf{Blue}\}^n \to {\mathbb R}$ then it’s best to just work abstractly with the orthogonal decomposition. However if there is a notion of, say, “addition” in $\Omega$ then there is a natural, canonical Fourier basis for $L^2(\Omega, \pi)$ when $\pi$ is the uniform distribution.
Continue reading §8.5: Abelian groups
Perhaps the most common generalized domain in analysis of boolean functions is the case of the hypercube with “biased” bits.
Continue reading §8.4: $p$biased analysis
In this section we describe a basisfree kind of “Fourier expansion” for functions on general product domains. We will refer to it as the orthogonal decomposition of $f \in L^2(\Omega^n, \pi^{\otimes n})$ though it goes by several other names in the literature: e.g., Hoeffding, Efron–Stein, or ANOVA decomposition.
Continue reading §8.3: Orthogonal decomposition
In this section we will revisit a number of combinatorial/probabilistic notions and show that for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, these notions have familiar Fourier formulas which don’t depend on the Fourier basis.
Continue reading §8.2: Generalized Fourier formulas


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