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Canonical static concept record

Convergence in probability

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Summary

A sequence $X_n$ converges in probability to $X$ when $\mathbb P(d(X_n,X)>\varepsilon)\to0$ for every $\varepsilon>0$ .

Record metadata

Canonical name: Convergence in probability
Concept ID: convergence_in_probability
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Probability
Domains: Probability, Measure, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

random variables $X_n,X$ with values in a metric space

Data

sequence $(X_n)$ and candidate limit $X$

Axioms / constraints

$\mathbb P(d(X_n,X)>\varepsilon)\to0$ for every $\varepsilon>0$

Notes

Convergence in probability is weaker than almost-sure convergence and stronger than convergence in distribution.

Concept sources

Wikipedia — Convergence of random variables — Convergence of random variables · encyclopedic reference · source ID wp-convergence-random-variables
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

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Law of large numbers Convergence in probability

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: law_large_numbers_to_probability_convergence
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: weak-law convergence

Authored explanation

The weak law asserts convergence in probability of sample averages to the relevant expectation under its stated hypotheses.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Law of large numbers — Law of large numbers · encyclopedic reference · source ID wp-law-large-numbers
Wikipedia — Convergence of random variables — Convergence of random variables · encyclopedic reference · source ID wp-convergence-random-variables
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

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No direct relations are authored in this direction.

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