Cauchy-Schwarz for random variables

Depends on

• Holder's inequality for random variables
• Cauchy-Schwarz inequality for $L^2$

Used by

• Autonomous all-moment Euler MLMC logarithmic cost Corollary
• The second-moment lower bound for positive probability Corollary
• An unbounded stopped exponential martingale needs uniform integrability Counterexample
• Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
• A noncommuting two-noise equation requires an area term Example
• A polynomial-tail law with finite matrix variance Example
• A square-integrable linear shift with nonsummable physical dependence Example
• A unit-root AR(1) has no weakly stationary solution with nonzero white noise Example
• An atomic sinusoid and an AR(1) process have different spectra Example
• Ito formula for Brownian powers Example
• Normal sample mean and residuals are independent Example
• One-step and multistep forecasts for a causal AR(1) Example
• Radial second moment of multidimensional Brownian motion Example
• Square of a martingale minus quadratic compensator Example
• Absolutely summable linear filters preserve weak stationarity Lemma
• Adjacent iid vector functionals admit a finite-moment strong approximation Lemma
• Characteristic exponential for a continuous local martingale with deterministic clock Lemma
• Continuous and strictly increasing norm distributions for density bridges Lemma
• Expected squared operator error of thresholded sparse covariance Lemma
• Finite-dependent vector central limit theorem Lemma
• Finite-horizon Brownian change of drift and relative entropy Lemma
• Gaussian mixture lower bound for weakly sparse covariance Lemma
• Gaussian orthogonality and the monomial expansion of the Hermite polynomials Lemma
• Independent centered Hilbert-valued elements have zero expected inner product Lemma
• Independent-tail replacement errors are decreasing and summability implies the rate condition Lemma
• Isometry for elementary predictable counting-process integrals Lemma
• Location-scale normal laws have their stated mean and variance Lemma
• Long-run covariance of an L2-m-approximable shift is trace class Lemma
• Martingale differences are orthogonal in l2 Lemma
• Oppositely monotone functions of independent coordinates have nonpositive covariance Lemma
• Pathwise relative-entropy error of the Langevin Euler interpolation Lemma
• Positive trace-class covariance defines a Hilbert Gaussian Lemma
• Strong-mixing covariance bound in terms of tail quantiles Lemma
• The Cameron–Martin unit ball is compact in the Banach norm Lemma
• The standard Gaussian law is determined by its moments Lemma
• Two-sided stationary extension preserves maximal-correlation bounds Lemma
• Uniform finite-horizon moments for tamed Euler approximations Lemma
• Uniform marginal-score bounds by derivative integration Lemma
• Weighted overlapping-block moment and covariance limits Lemma
• Zero-variance geometric Markov sums have a uniform strong rate Lemma

…and 36 more results.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. R. Norris, Probability and Measure, Theorem 4.4.1 (standard reference, not scraped)