TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Chebyshev weak law for uncorrelated arrays

Statement

For each $n \geq 1$ , let $X_{n, 1}, \dots, X_{n, r_{n}}$ be square-integrable real random variables on one probability space, pairwise uncorrelated within the row, where $r_{n} \geq 0$ is finite. Set $S_{n} = \sum_{k = 1}^{r_{n}} X_{n, k}$ and let $b_{n} > 0$ be deterministic. If $v_{n} : = b_{n}^{- 2} \sum_{k = 1}^{r_{n}} Var (X_{n, k}) ⟶ 0,$ then $(S_{n} - E S_{n}) / b_{n} \to 0$ in $L^{2}$ and in probability. More precisely, its second moment is $v_{n}$ , and its probability of absolute value at least $ε > 0$ is at most $v_{n} / ε^{2}$ . No independence between rows is required.

Facts & Assumptions

[F1]

Variance and covariance identities for random variables: Let $X, Y$ be square-integrable real random variables on one probability space. Then $Var (X) = E [X^{2}] - E [X]^{2},$ $Cov (X, Y) = E [X Y] - E [X] E [Y] .$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

[F2]

Chebyshev's inequality for random variables: If $X$ is a square-integrable real random variable and $a > 0$ , then $P (∣ X - E [X] ∣ \geq a) \leq \frac{Var (X)}{a^{2}} .$

[F3]

Convergence in probability: For real random variables $(X_{n})$ and $X$ on one probability space, write $X_{n} \to X$ in probability when, for every $ε > 0$ , $P (∣ X_{n} - X ∣ > ε) ⟶ 0.$ This is precisely def-convergence-in-measure for the probability measure.

[F4]

$L^{p}$ convergence for random variables: Let $1 \leq p \leq \infty$ . For real random variables whose classes lie in $L^{p} (P)$ as defined by def-l-p-space-as-a-quotient-by-null-functions, write $X_{n} \to X$ in $L^{p}$ when $∥ [X_{n}] - [X] ∥_{L^{p} (P)} ⟶ 0.$ For $p < \infty$ , this norm is $∥ [X_{n}] - [X] ∥_{L^{p} (P)} = (E ∣ X_{n} - X ∣^{p})^{1 / p};$ for $p = \infty$ , it is the essential-supremum norm. Thus the assertion concerns almost-everywhere equivalence classes, not chosen representatives.

[F5]

Linearity, monotonicity, and the modulus bound for expectation: Let $X, Y$ be integrable real or complex random variables on one probability space. 1. For scalars $a, b$ , $E [a X + b Y] = a E [X] + b E [Y] .$ 2. If $X$ and $Y$ are real-valued and $X \leq Y$ almost surely, then $E [X] \leq E [Y] .$ 3. $∣ E [X] ∣ \leq E [∣ X ∣] .$

Proof

Given: The objects and hypotheses of the statement.

1.1F5givenalgebra

Put $Z_{n} = b_{n}^{- 1} \sum_{k} (X_{n, k} - E X_{n, k}) = (S_{n} - E S_{n}) / b_{n}$ . Finite linearity gives $E Z_{n} = 0$ . Square integrability of the finite sum follows from $(\sum_{k = 1}^{r} z_{k})^{2} \leq r \sum_{k} z_{k}^{2}$ for $r \geq 1$ ; the empty sum is zero.

2.1F1F4step 1.1algebra

Covariance bilinearity and the zero off-diagonal covariances give $E ∣ Z_{n} ∣^{2} = b_{n}^{- 2} \sum_{k} Var (X_{n, k}) = v_{n}$ . This includes a singleton row and zero variances without division by a variance. Thus the $L^{2}$ norm tends to zero.

3.1F2F3step 2.1∎

For every $ε > 0$ , Chebyshev gives $P (∣ Z_{n} ∣ \geq ε) \leq v_{n} / ε^{2} \to 0$ . This also bounds the strict event defining convergence in probability.

Depends on

Used by

IID finite-variance weak law Corollary
A macroscopic row term defeats averaging Counterexample
A nonidentical Bernoulli weak law Example
Truncation weak law for independent arrays Theorem

Dependency tree · two levels

23 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

Theorems 2.2.1, 2.2.3 and 2.2.6, pp. 56–59 (standard reference, not scraped)