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

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Convergence in probability implies convergence in distribution

Statement

If $X_{n} \to X$ in probability, then $X_{n} \Rightarrow X$ .

Facts & Assumptions

Given: $X_{n} \to X$ in probability.

[L1]

Distributional convergence is CDF convergence at every continuity point of the limit CDF (Convergence in distribution for real random variables).

[L2]

Convergence in probability controls every fixed error threshold (Convergence in probability).

Proof

technique · direct

1.1

Fix a continuity point $x$ of $F_{X}$ and $δ > 0$ . The following inclusions give a CDF squeeze. [given] ${X \leq x - δ} ∖ {∣ X_{n} - X ∣ > δ} \subseteq {X_{n} \leq x} \subseteq {X \leq x + δ} \cup {∣ X_{n} - X ∣ > δ}$ give $F_{X} (x - δ) - p_{n} \leq F_{X_{n}} (x) \leq F_{X} (x + δ) + p_{n},$ where $p_{n} = P (∣ X_{n} - X ∣ > δ)$ .

given

2.1

By [L2], $p_{n} \to 0$ ; taking liminf and limsup in step 1.1 gives the required limiting bounds. [step 1.1, L1, L2] $δ ↓ 0$ uses continuity at $x$ to give $F_{X_{n}} (x) \to F_{X} (x)$ . By [L1], this is $X_{n} \Rightarrow X$ .

step 1.1L1L2∎

Depends on

Used by

Complete convergence implication diagram Remark

Dependency tree · two levels

9 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)