283 Boolean Analysis — Moment Bounds
283.1 Overview
Reasonability compares fourth and second moments of a real random variable. Fourth-moment tail estimates, finite-atom bounds, and Paley–Zygmund give quantitative upper-tail and nonvanishing probabilities.
283.2 Declarations
Let \(X\colon \Omega \to \mathbb {R}\) be a function on a measurable space, let \(P\) be a measure on \(\Omega \), and let \(B\in \mathbb {R}\). The function \(X\) is \(B\)-reasonable under \(P\) when \(P\) is a probability measure, \(B\ge 1\), \(X\) is almost everywhere measurable with finite fourth absolute moment, and
Let \(X\in L^4\) on a probability space, let \(B\ge 1\), and suppose \(\mathbb {E}X^4\le B(\mathbb {E}X^2)^2\), \(X^4\) is integrable, and \(\mathbb {E}X^2\gt 0\). For every \(t\gt 0\), \(\Pr [X^4\ge t^4(\mathbb {E}X^2)^2]\le B/t^4\).
Let \(\Omega \) be a finite sample space with all subsets measurable, let \(\pi \) be a probability mass function on \(\Omega \), and give \(\Omega \) the probability measure \(P(\{ \omega \} )=\pi (\omega )\). For any function \(X\colon \Omega \to \mathbb {R}\), if \(\mu \gt 0\) satisfies \(\mu \le \pi (\omega )\) for every \(\omega \in \Omega \), then \(X\) is \((1/\mu )\)-reasonable under \(P\); in particular,
Let \(Z\) be a measurable real-valued random variable on a probability space that is almost surely nonnegative and has finite first and second moments \(\mathbb {E}[Z]\) and \(\mathbb {E}[Z^2]\), with \(\mathbb {E}[Z] \gt 0\). Then for every \(\theta \) with \(0 \le \theta \le 1\),
Let \(X\) be measurable on a probability space, assume \(X\in L^4\), \(X^2\) and \(X^4\) are integrable, and \(\mathbb {E}X^2\gt 0\). If \(B\ge 1\) and \(\mathbb {E}X^4\le B(\mathbb {E}X^2)^2\), then for every \(0\le t\le 1\), \(\Pr [X^2\gt t^2\mathbb {E}X^2]\ge (1-t^2)^2/B\).