TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Convergence in distribution to a constant is convergence in probability

Statement

If the real random variables $X_{n}$ are defined on one probability space and $X_{n} \Rightarrow c$ for a real constant $c$ , then $X_{n} \to c$ in probability on that space.

Facts & Assumptions

Given: Real random variables $X_{n}$ on one probability space, $X_{n} \Rightarrow c$ , and $ε > 0$ .

[L1]

Distributional convergence gives CDF convergence at continuity points (Convergence in distribution for real random variables).

Proof

technique · direct

1.1L1

The constant-law CDF is continuous at $c - ε$ and $c + ε$ , so [L1] gives [L1] $F_{X_{n}} (c - ε) \to 0$ and $F_{X_{n}} (c + ε) \to 1$ .

2.1step 1.1∎

The error event is contained in the following union. [step 1.1] ${X_{n} \leq c - ε} \cup {X_{n} > c + ε}$ , its probability is at most $F_{X_{n}} (c - ε) + 1 - F_{X_{n}} (c + ε)$ , which tends to zero.

Depends on

Used by

Central-limit convergence is only in distribution Remark

Dependency tree · two levels

8 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 3.2 (standard reference, not scraped)