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Linearity, monotonicity, and the modulus bound for expectation
Statement
Let be integrable real or complex random variables on one probability space.
- For scalars ,
- If and are real-valued and almost surely, then
Facts & Assumptions
Given: Integrable random variables .
Expectation is the Lebesgue integral against the probability measure (Expectation of a nonnegative or integrable random variable).
The Lebesgue integral is linear on , the nonnegative integral is monotone, and the modulus of an integral is bounded by the integral of the modulus (The Lebesgue integral is linear on , Monotonicity and nonnegative homogeneity of the nonnegative integral, The modulus of an integral is bounded by the integral of the modulus).
Almost-everywhere equal integrable functions have equal integrals on every measurable event. (Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree)
The positive part of a measurable real function is measurable. (Closure properties of measurable functions used by the integral)
Proof
Rewriting expectation as the integral by [L1], linearity in [L2] gives
Applying the integral triangle inequality from [L2] after [L1] gives
If almost surely, put , which is integrable by [L2]. Its positive part is measurable by [L4] and integrable because . Since almost surely, almost surely, so [L3] gives , where the last inequality concerns a pointwise nonnegative integrand. By [L1] and step 1.1, , so .
Steps 1.1, 1.2, and 2.1 prove the three assertions.
Depends on
- Expectation of a nonnegative or integrable random variable
- The Lebesgue integral is linear on $L^1(\mu)$
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The modulus of an integral is bounded by the integral of the modulus
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Closure properties of measurable functions used by the integral
Used by
- A curved normal family can make a sufficient statistic incomplete Counterexample
- Collinear columns identify a fitted mean, not its coefficients Counterexample
- Unequal error variances can make OLS fail to be BLUE Counterexample
- Fixed-design linear regression model Definition
- A Gaussian maximum without independence Example
- A Rademacher sum Example
- An atomic sinusoid and an AR(1) process have different spectra Example
- Common random numbers can remove finite-difference noise Example
- Constrained least squares in a two-coordinate orthogonal sequence model Example
- Gaussian reparameterization gives two gradient estimators Example
- Noisy simulator output separates intrinsic noise from input variance Example
- A Bernoulli(p) variable has mean p and variance p(1-p); a binomial(n,p) variable has mean np and variance np(1-p) Lemma
- A PRDS bound for each BH true-null contribution Lemma
- Basic finite covariance matrix identities for robust mean bounds Lemma
- Bounded centered convergent series have summable variances Lemma
- Continuous and strictly increasing norm distributions for density bridges Lemma
- Expectation, variance, and MSE of a finite MLMC estimator Lemma
- Finite-horizon crossing bound for a nonnegative test supermartingale Lemma
- Gaussian orthogonality and the monomial expansion of the Hermite polynomials Lemma
- Location-scale normal laws have their stated mean and variance Lemma
- One-sided concentration bound for independent median trials Lemma
- Oppositely monotone functions of independent coordinates have nonpositive covariance Lemma
- Products of sub-Gaussian coordinates are sub-exponential Lemma
- Products of variables with Gaussian tails are sub-exponential Lemma
- The Hoeffding MGF bound for a variable in a unit interval Lemma
- The standard Gaussian law is determined by its moments Lemma
- Variance and covariance identities for random variables Lemma
- A dominated pathwise simulation gradient is unbiased Theorem
- A finite stochastic adjoint gives the discrete gradient under domination Theorem
- A proper Lyapunov bound prevents reaction-network explosion Theorem
- An independent pilot coefficient preserves conditional unbiasedness Theorem
- Antithetic variance and a monotone sufficient condition Theorem
- Asymptotic unbiasedness of lattice covariance and variogram estimators Theorem
- Benjamini–Hochberg FDR under independence Theorem
- Bernstein inequality for bounded independent sums Theorem
- BH controls FDR under PRDS Theorem
- BY controls FDR under arbitrary dependence Theorem
- Chebyshev weak law for uncorrelated arrays Theorem
- Classical bounded-sample mean baseline Theorem
- Compensators, brackets, and jump-integral isometry for reaction counts Theorem
…and 33 more results.
Dependency tree · two levels
19 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 (standard reference, not scraped)