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Almost-sure convergence of an independent series is a zero-one event

Statement

Let $(X_{n})_{n \in N}$ be an independent sequence of real random variables. Then the event

${\sum_{n = 0}^{\infty} X_{n} converges}$

has probability $0$ or $1$ .

Facts & Assumptions

Given: An independent sequence of real random variables $(X_{n})_{n \in N}$ .

[L1]

The tail sigma-algebra consists of the events determined by all but finitely many coordinates. (Tail sigma-algebra of a sequence)

[L2]

Finite sums and absolute values of measurable real-valued functions are measurable. (Arithmetic and lattice operations preserve measurability whenever they are defined)

[L3]

Every tail event of an independent sequence has probability $0$ or $1$ . (Kolmogorov zero-one law)

Proof

technique · direct

1.1

Let $E : = {\sum_{n = 0}^{\infty} X_{n} converges}$ and fix $m \in N$ . The series $\sum_{n = 0}^{\infty} X_{n}$ converges if and only if the tail series $\sum_{n = m}^{\infty} X_{n}$ converges, because removing finitely many initial terms changes every partial sum by a fixed finite constant. By the Cauchy criterion, $E = ⋂_{r = 1}^{\infty} ⋃_{N = m}^{\infty} ⋂_{q \geq p \geq N} {∣ \sum_{n = p}^{q} X_{n} ∣ \leq \frac{1}{r}} .$ For $q \geq p \geq N \geq m$ , the partial sum $\sum_{n = p}^{q} X_{n}$ is measurable with respect to $σ (X_{n} : n \geq m)$ by repeated use of [L2], so each displayed event lies in $σ (X_{n} : n \geq m)$ . Therefore $E \in σ (X_{n} : n \geq m)$ for every $m$ .

givenL1L2

2.1

Step 1.1 shows that $E$ lies in the tail sigma-algebra, so [L3] gives $P (E) \in {0, 1}$ .

step 1.1L3∎

Depends on

Used by

Convergence of an independent random series is a zero-one event Example

Dependency tree · two levels

12 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

S. R. S. Varadhan, Probability Theory, Theorem 3.15 (standard reference, not scraped)