Convergence in distribution

Mathematical statement

A common probability space or chosen coupling is not required. A finite prefix does not prove convergence, and the definition imposes no convergence requirement at discontinuities of the limiting CDF. Convergence in distribution does not generally imply convergence in probability, almost surely or in Lp.

Representative formula (exact TeX)

D(X)

Assumptions

• This slice uses a symbolic Bernoulli-parameter tail p_n=p+c/(n+s), restricted to [0,1].
• Checks distinguish continuity points of the limiting CDF and the explicitly selected thresholds.

Symbol scope

• D — convergence-in-distribution record
• F — CDF

Intuition, definition, and misconceptions

Only continuity points of the limiting CDF carry the defining requirement.

A common coupling is not required by the definition.

An explicit symbolic parameter tail supplies a universal certificate.

A finite selection of n does not prove convergence, jump points need not satisfy the defining requirement, and convergence in probability cannot be inferred automatically.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Cumulative distribution function — The definition uses continuity points of the limiting CDF.
• Function limits on punctured real domains — Each required point has a pointwise-limit obligation.
• Sequence — The CDFs F_n are indexed by n.
• Finite joint distribution — Discrete laws provide the CDFs.

Verification boundary

Only the explicit symbolic Bernoulli family is handled; stronger convergence modes are not inferred.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add the Portmanteau theorem and central limit theorem.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.