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Chebyshev's inequality for random variables
Statement
If is a square-integrable real random variable and , then
Facts & Assumptions
Given: A square-integrable real random variable and a real number .
Variance is the expectation of the squared centered variable (Moments, variance, and covariance on a probability space).
Markov's inequality applies to every nonnegative random variable (Markov's inequality for random variables).
Proof
The random variable is nonnegative, and
Applying [L2] to and using [L1] gives
Depends on
Used by
- Brownian paths have infinite total variation Corollary
- The excess-tail functional cannot upper-bound the trimmed mean Counterexample
- Weak tests against likelihood tilts do not ensure almost-sure posterior consistency Counterexample
- A square-integrable linear shift with nonsummable physical dependence Example
- Markov and Chebyshev sharpness Example
- Weakly close root laws can still give failed uniform coverage Example
- Bernoulli root laws are uniformly close in Levy–Prokhorov distance Lemma
- Conditional multinomial resampling central limit theorem Lemma
- Conditional mutation and weighted correction CLT Lemma
- Conditional residual resampling central limit theorem away from integer weights Lemma
- Finite-dependent vector central limit theorem Lemma
- Finite-horizon SMC empirical laws converge for integrable tests Lemma
- KL prior mass gives a posterior denominator lower bound in probability Lemma
- Local KL and log-square mass bounds a non-IID posterior denominator Lemma
- Posterior radius, bias and noise comparisons for empirical Bayes Lemma
- Scaled Plancherel profile moments converge in probability Proposition
- The general inequalities compare cleanly with the published finite ones Remark
- Brownian quadratic variation on dyadic partitions Theorem
- Chebyshev weak law for uncorrelated arrays Theorem
- Classical bounded-sample mean baseline Theorem
- Etemadi strong law for pairwise independent iid variables Theorem
- m-out-of-n bootstrap and subsampling recover the uniform endpoint root Theorem
- Pairwise-independent Borel-Cantelli frequency law Theorem
- Quadratic covariation of Brownian Ito processes Theorem
- Quadratic variation of an Ito integral Theorem
- Same-sample coefficient remainder is of order O_P(n⁻¹) Theorem
- Studentized Wilcoxon validity under heterogeneous laws Theorem
- Variance-aware quantum Monte Carlo guarantee Theorem
Dependency tree · two levels
4 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6.1 (standard reference, not scraped)
- J. R. Norris, Probability and Measure, Section 4.2 (standard reference, not scraped)