Probability theory and stochastic processes Modes of convergence Concept
Convergence of random variables
Builds on Almost sure convergenceConvergence in probabilityConvergence in distributionConvergence in mean
Summary
A sequence of random variables can converge to a limit in several different senses. The main modes are almost sure convergence, convergence in probability, convergence in mean and convergence in distribution. Almost sure convergence and convergence in mean each imply convergence in probability, and convergence in probability implies convergence in distribution. The converse implications are false in general.
Remarks
Convergence in distribution concerns only the laws of the random variables. So the random variables can be on different probability spaces. The law of large numbers is a statement about convergence in probability or almost surely. The central limit theorem is a statement about convergence in distribution.
Reading path
To understand this page from the ground up, read these 51 pages in order. The foundations come first.
- Construction of the reals from Cauchy sequences concept
- Element definition
- Integers object
- Ordered field definition
- Proposition concept
- Random experiment concept
- Rational numbers object
- Real numbers object
- Set definition
- Ring of integers statement
- Disjunction definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Union definition
- Sample space definition
- The rationals form a field statement
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- The rationals form an ordered field statement
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Order on the reals definition
- Power set definition
- Complement definition
- Function definition
- Interval definition
- Injective function definition
- Preimage of a set definition
- Sigma-algebra definition
- Countable set definition
- Event definition
- Borel sigma-algebra definition
- Probability axioms concept
- Probability measure definition
- Probability distribution definition
- Probability space definition
- Almost surely definition
- Random variable definition
- Almost sure convergence definition
- Cumulative distribution function definition
- Discrete random variable definition
- Convergence in probability definition
- Random variable with a density definition
- Convergence in distribution definition
- Expected value definition
- Convergence in mean definition
- Convergence of random variables concept
Where this leads
2 pages build on this one.
Further reading
About this page
| Tag | 0FQ968YKA1 |
|---|---|
| URI | m2://0FQ968YKA1 |
| Aliases | modes-of-convergence-of-random-variables, stochastic-convergence |
| Kind | concept |
| Writing | W0 stub |
| MSC | 60F05, 60F15 |
| Uses | Almost sure convergence, Convergence in probability, Convergence in distribution, Convergence in mean |
| Same as |
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