ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The three series impose separate conditions

Example

Assume countable choice and dependent choice. At cutoff $A = 1$ , each of the following independent-sequence constructions violates exactly one of the three-series conditions:

For $n \geq 2$ , let $X_{n} = 2$ with probability $1 / n$ and $X_{n} = 0$ otherwise; set $X_{1} = 0$ . Only the large-jump probability series diverges.
Let $X_{n} = 1 / n$ deterministically. Only the truncated mean series diverges.
Let $X_{n} = ϵ_{n} / \sqrt{n}$ for independent fair signs. Only the truncated variance series diverges.

None of these series converges almost surely.

Facts & Assumptions

[F1]

Kolmogorov three-series theorem: Let $(X_{n})_{n \geq 1}$ be independent real random variables and fix $A > 0$ . Put $Y_{n} = X_{n} 1_{{∣ X_{n} ∣ \leq A}}$ . Then $\sum_{n} X_{n}$ converges almost surely if and only if all three conditions hold: $\sum_{n} P (∣ X_{n} ∣ > A) < \infty, \sum_{n} E Y_{n} converges in R, \sum_{n} Var (Y_{n}) < \infty .$ The conditions hold for some $A > 0$ if and only if they hold for every $A > 0$ . No moment assumption is imposed on the untruncated variables.

[F2]

Second Borel-Cantelli lemma under pairwise independence: Let $(A_{n})_{n \in N}$ be pairwise independent events with $\sum_{n = 0}^{\infty} P (A_{n}) = + \infty .$ Then $P (A_{n} i . o .) = 1.$

[F3]

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$ , $\sum_{k \geq 1} \frac{1}{k^{p}} converges ⟺ p > 1.$

[F4]

Coordinate random elements of a countable product are independent: Under the measure of thm-countable-product-of-probability-spaces, the coordinate maps $X_{n} (x) = x_{n}$ have laws $μ_{n}$ and are independent.

[F5]

Assuming countable and dependent choice, countable products of arbitrary probability spaces: Assume countable choice and dependent choice. For probability spaces $(E_{n}, E_{n}, μ_{n})_{n \in N}$ there is a unique probability measure $μ$ on $C_{N}$ such that, for every finite $F$ , its $F$ -coordinate marginal is $⨂_{n \in F} μ_{n}$ .

Verification

Given: The construction and assumptions above.

1.1

Under countable choice and dependent choice, the countable product of the stated finite probability spaces constructs the first and third independent sequences; deterministic coordinates construct the second. In the first construction the zero truncations at $1$ are all zero, so their mean and variance series vanish, but $\sum_{n \geq 2} P (∣ X_{n} ∣ > 1) = \sum_{n \geq 2} 1 / n = \infty$ . The second Borel–Cantelli lemma gives infinitely many terms equal to $2$ almost surely; hence the terms fail to tend to zero.

F5F4F3F2given

1.2

For the second construction, every term is retained at $A = 1$ , its variance is zero, and there are no large jumps. Its truncated mean series is $\sum_{n} 1 / n = \infty$ . Thus exactly the mean condition fails and its deterministic partial sums diverge.

F3givenalgebra

2.1

For the third construction every term, including $n = 1$ , is retained, there are no large jumps, and the means vanish. Its variance series is $\sum_{n} 1 / n = \infty$ . The three-series theorem rules out almost-sure convergence. Each construction therefore isolates exactly the claimed failed condition.

F3F1givenalgebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Theorem 3.12, pp. 66–68, direct specializations (standard reference, not scraped)
Example 2.5.7, p. 85, variance obstruction (standard reference, not scraped)