# Proofs of convergence of random variables

[Convergence of random variables](https://www.edgepedia.org/convergence-of-random-variables) is the family of mathematical statements describing how a sequence \( X_1, X_2, \ldots \) of random variables approaches a limiting random variable \( X \), and the proofs that establish which of these statements imply which. [Probability theory](https://www.edgepedia.org/probability-theory) uses four standard modes of convergence: almost sure convergence, convergence in probability, convergence in \( L^p \), and convergence in distribution<sup>[1](https://www.theorempath.com/topics/modes-of-convergence-random-variables)</sup>. The subject is central to probability theory and its applications to statistics and stochastic processes<sup>[2](https://onlinelibrary.wiley.com/doi/10.1002/9781118593103.ch11)</sup>.

| Key fact | Detail |
|---|---|
| Four modes | Almost sure, in probability, in \( L^p \) (for \( p \geq 1 \)), and in distribution<sup>[1](https://www.theorempath.com/topics/modes-of-convergence-random-variables)</sup> |
| Definition of convergence in probability | \( X_n \xrightarrow{P} X \) if for every \( \varepsilon > 0 \), \( \lim_{n \to \infty} P(|X_n - X| \geq \varepsilon) = 0 \)<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup> |
| Definition of \( L^p \) convergence | \( X_n \xrightarrow{L^p} X \) if \( E|X|^p < \infty \) and \( E|X_n - X|^p \to 0 \)<sup>[4](https://www.statslab.cam.ac.uk/~mike/probability/chapter1-2.pdf)</sup> |
| Strict hierarchy | Almost sure \( \Rightarrow \) in probability \( \Leftarrow \) in \( L^p \); all three \( \Rightarrow \) in distribution, and each reverse implication fails<sup>[5](https://math.stackexchange.com/questions/1170559/convergence-types-in-probability-theory-counterexamples)</sup> |
| Borel–Cantelli criterion | If \( \sum_{n \geq 1} P(|X_n - X| \geq \varepsilon) < \infty \) for every \( \varepsilon > 0 \), then \( X_n \to X \) almost surely<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup> |
| Uniform integrability bridge | If \( X_n \xrightarrow{P} X \), then \( X_n \xrightarrow{L^1} X \) if and only if \( (X_n) \) is uniformly integrable<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup> |
| Lévy continuity theorem | Convergence in distribution is equivalent to pointwise convergence of characteristic functions to a limit continuous at the origin<sup>[6](https://www.math.stonybrook.edu/~aknapp/download/a2-realanal-inside.pdf)</sup> |

## The four modes and their definitions

[Almost sure convergence](https://www.edgepedia.org/almost-sure-convergence) means that \( X_n(\omega) \to X(\omega) \) for all outcomes \( \omega \) outside a set of probability zero; in the language of general measure theory this is almost everywhere convergence, and when the measure is a probability it is called almost sure convergence<sup>[7](https://www.sciencedirect.com/topics/mathematics/almost-sure-convergence)</sup>. [Convergence in probability](https://www.edgepedia.org/convergence-in-probability) quantifies the shrinking probability of large deviations: for every \( \varepsilon > 0 \), the probability of the event \( |X_n - X| \geq \varepsilon \) tends to zero<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>. Convergence in \( L^p \), for \( p \geq 1 \), requires \( E|X|^p < \infty \) and \( E|X_n - X|^p \to 0 \)<sup>[4](https://www.statslab.cam.ac.uk/~mike/probability/chapter1-2.pdf)</sup>.

[Convergence in distribution](https://www.edgepedia.org/convergence-in-distribution) is different in kind. It requires only that the distribution functions converge pointwise at every continuity point of the limit, so the \( X_n \) and \( X \) need not be defined on the same probability space and need not be pointwise close<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>. Lecture notes often group the first three as "spatial" convergence, describing closeness of \( X_n \) and \( X \) on a common space, and convergence in distribution as "measure" convergence, describing similarity of probabilistic behavior<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

## The hierarchy of implications

The valid implications form a strict hierarchy. Almost sure convergence implies convergence in probability, and convergence in \( L^p \) also implies convergence in probability; both of these imply convergence in distribution<sup>[5](https://math.stackexchange.com/questions/1170559/convergence-types-in-probability-theory-counterexamples)</sup>. The proof that \( L^p \) convergence implies convergence in probability comes from integrating the pointwise inequality \( \varepsilon^p 1_{\{|X_n - X| > \varepsilon\}} \leq |X_n - X|^p \), which gives \( P(|X_n - X| > \varepsilon) \leq E|X_n - X|^p / \varepsilon^p \to 0 \)<sup>[8](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)</sup>. The proof that almost sure convergence implies convergence in probability applies the dominated convergence theorem to the indicators \( 1_{\{|X_n - X| \geq \varepsilon\}} \), which converge almost surely to zero<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

Almost sure convergence and \( L^p \) convergence are generally not compatible: neither implies the other<sup>[2](https://onlinelibrary.wiley.com/doi/10.1002/9781118593103.ch11)</sup>. Since convergence in probability does not imply \( L^p \) convergence, almost sure convergence cannot imply it either<sup>[8](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)</sup>.

## Counterexamples separating the modes

**Convergence in probability does not imply almost sure convergence.** On the probability space \( (0,1] \) with [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure), define a sequence of indicator functions of intervals that march across the unit interval and shrink: \( X_1 = 1_{(1/2,1]} \), \( X_2 = 1_{(0,1/2]} \), \( X_3 = 1_{(3/4,1]} \), \( X_4 = 1_{(1/2,3/4]} \), and so on<sup>[5](https://math.stackexchange.com/questions/1170559/convergence-types-in-probability-theory-counterexamples)</sup>. For every \( \varepsilon > 0 \), \( P(|X_n| \geq \varepsilon) \leq 2^{-k} \to 0 \) within each block, so \( X_n \xrightarrow{P} 0 \); but every \( \omega \in [0,1] \) has \( X_n(\omega) = 1 \) for infinitely many \( n \), so the sequence fails to converge almost surely<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

**Convergence in \( L^p \) does not imply almost sure convergence.** Let \( X_n \) be independent random variables taking the value 1 with probability \( 1/n \) and 0 with probability \( 1 - 1/n \). Then \( X_n \to 0 \) in probability but not almost surely, by the second [Borel–Cantelli lemma](https://www.edgepedia.org/borel-cantelli-lemma)<sup>[8](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)</sup>.

**Convergence in probability does not imply \( L^p \) convergence**, not even when all variables lie in \( L^p \). On \( (0,1) \) with Lebesgue measure, take \( X_n = n^{1/p} 1_{(0,1/n)} \). Then \( X_n \to 0 \) in probability but not in \( L^p \)<sup>[8](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)</sup>.

**Convergence in distribution does not imply convergence in probability.** Take any two random variables \( X \) and \( Y \) with \( X \neq Y \) almost surely but \( X = Y \) in distribution; the constant sequences \( X_n = X \) converge in distribution but not in probability<sup>[5](https://math.stackexchange.com/questions/1170559/convergence-types-in-probability-theory-counterexamples)</sup>. A concrete version: if \( X \sim \mathrm{Bern}(1/2) \) and \( Y_n = 1 - X \) for every \( n \), then \( Y_n \xrightarrow{d} 1 - X \), which has the same law as \( X \), yet \( X \neq 1 - X \) almost surely<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

## Proof tools for almost sure and Lp convergence

**The Borel–Cantelli criterion.** A practical sufficient condition for almost sure convergence is summability of tail probabilities: if \( \sum_{n \geq 1} P(|X_n - X| \geq \varepsilon) < \infty \) for every \( \varepsilon > 0 \), then \( X_n \to X \) almost surely. The proof applies the first Borel–Cantelli lemma to show that for each \( k \), the event \( |X_n - X| \geq 1/k \) occurs only finitely many times almost surely<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

**The subsequence principle.** Convergence in probability guarantees an almost surely convergent subsequence: from \( X_n \xrightarrow{P} X \) one constructs inductively integers \( n_k \) with \( P(|X_{n_k} - X| > 2^{-k}) \leq 2^{-k} \), and the first Borel–Cantelli lemma makes \( X_{n_k} \to X \) almost surely<sup>[8](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)</sup>. This also underlies a uniqueness argument: if \( X_n \xrightarrow{P} X \) and \( X_n \xrightarrow{P} Y \), then \( X = Y \) almost surely<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

**Uniform integrability.** The bridge from convergence in probability to \( L^1 \) convergence is uniform integrability. If \( X_n \xrightarrow{P} X \), then \( X_n \xrightarrow{L^1} X \) if and only if the family \( (X_n) \) is uniformly integrable; in measure theory this is the Vitali convergence theorem in its finite-measure case<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>. Two standard sufficient conditions are domination by an integrable random variable \( Z \) and a moment criterion: \( \sup_i E(|X_i|^p) < \infty \) for some \( p > 1 \) implies uniform integrability<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>. Under uniform integrability and almost sure convergence, expectations pass to the limit, \( E(X_n) \to E(X) \), a corollary that generalizes the dominated convergence theorem<sup>[3](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)</sup>.

## Proof tools for convergence in distribution

**Characteristic functions.** The Lévy continuity theorem states that \( X_n \) converges in distribution to \( X \) if and only if the characteristic functions \( \varphi_{X_n}(t) \) converge pointwise to \( \varphi_X(t) \) for every \( t \); more generally, if \( \varphi_{X_n}(t) \to \varphi(t) \) for all \( t \) and \( \varphi \) is continuous at 0, then \( \varphi \) is a characteristic function and the \( X_n \) converge in distribution to the corresponding law<sup>[6](https://www.math.stonybrook.edu/~aknapp/download/a2-realanal-inside.pdf)</sup>. This reduces convergence in distribution, which is awkward to handle directly, to pointwise convergence of Fourier transforms, and it is the standard route to proving the central limit theorem<sup>[9](https://www.math.hkust.edu.hk/~makchen/MATH5411/Chap2Sec1.pdf)</sup>.

**The Portmanteau theorem and Skorokhod representation.** Weak convergence of probability measures on a metric space is characterized by several equivalent conditions: convergence of integrals against all bounded continuous functions, bounds on limsup over closed sets and liminf over open sets, and convergence on sets whose boundary has limiting measure zero<sup>[10](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup>. One equivalent condition is the Skorokhod representation theorem: there exist random variables \( \hat{X}_n \) with the laws of \( X_n \) that converge almost surely on a common space<sup>[10](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup>.

**Continuous mapping and Slutsky.** If \( X_n \xrightarrow{d} X \) and \( f \) is continuous, then \( f(X_n) \xrightarrow{d} f(X) \)<sup>[10](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup>. Slutsky's theorem combines sequences: if \( X_n \xrightarrow{d} Z \) and \( Y_n \xrightarrow{P} c \) for a constant \( c \), then \( X_n + Y_n \xrightarrow{d} Z + c \) and \( X_n Y_n \xrightarrow{d} cZ \), conclusions drawn without any knowledge of the joint distribution of the two sequences<sup>[11](https://www.emerald.com/books/oa-edited-volume/20922/chapter/107209997/Stochastic-Convergence)</sup>.

## Applications

The modes of convergence carry the classical limit theorems. The weak law of large numbers is a statement of convergence in probability, the strong law a statement of almost sure convergence; the two laws have the same hypotheses and the same limit and differ only in the mode of convergence<sup>[12](https://youyuanwu.github.io/misc-docs/math/11-probability/06-convergence-modes.html)</sup>.

In statistics, an estimator \( T_n \) of a parameter \( \theta \) is defined to be consistent when \( T_n \xrightarrow{P} \theta \); the weak law makes the sample mean a consistent estimator of the population mean, and the continuous mapping theorem transfers this to functions of it, for example the sample variance converging in probability to \( \sigma^2 \)<sup>[13](https://www.stat.purdue.edu/~mlevins/docs/stat516/lecture_notes/Lecture9_STAT517.pdf)</sup>.

In the numerical solution of stochastic differential equations, the same distinction reappears as strong versus weak convergence: strong schemes aim to track individual sample paths accurately, while weak schemes target distributional features such as expectations of functionals<sup>[14](https://www.nature.com/nature-index/topics/l4/numerical-methods-for-stochastic-differential-equations)</sup>. The [Euler–Maruyama method](https://www.edgepedia.org/euler-maruyama-method) converges with strong order \( 1/2 \) and weak order 1, and Milstein's method adds a correction term to reach strong order 1<sup>[15](https://epubs.siam.org/doi/full/10.1137/S0036144500378302)</sup>.

## References

1. [Modes of Convergence of Random Variables, TheoremPath](https://www.theorempath.com/topics/modes-of-convergence-random-variables)
2. [Convergence Types. Almost Sure Convergence. Lp-Convergence. Convergence in Probability, Handbook of Probability (Florescu & Tudor), Wiley](https://onlinelibrary.wiley.com/doi/10.1002/9781118593103.ch11)
3. [Probability Theory lecture notes, ETH Zürich](https://metaphor.ethz.ch/x/2025/hs/401-3601-00L/main.pdf)
4. [Introduction to Probability, Chapter 1–2, University of Cambridge Stats Lab](https://www.statslab.cam.ac.uk/~mike/probability/chapter1-2.pdf)
5. [Convergence types in probability theory: Counterexamples, Mathematics Stack Exchange](https://math.stackexchange.com/questions/1170559/convergence-types-in-probability-theory-counterexamples)
6. [Lévy Continuity Theorem, Stony Brook real analysis notes](https://www.math.stonybrook.edu/~aknapp/download/a2-realanal-inside.pdf)
7. [Almost Sure Convergence, ScienceDirect Topics (Roussas, An Introduction to Measure-Theoretic Probability)](https://www.sciencedirect.com/topics/mathematics/almost-sure-convergence)
8. [Convergences in Probability, lecture notes by Davide Giraudo, Université de Strasbourg](https://irma.math.unistra.fr/~giraudo/SPbU_Lecture_Notes_Convergences_in_Probability_Davide_Giraudo.pdf)
9. [Central limit, Chapter 2, MATH 5411 notes, HKUST](https://www.math.hkust.edu.hk/~makchen/MATH5411/Chap2Sec1.pdf)
10. [Probability Theory lecture notes (Chewi), UC Berkeley 205B](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)
11. [Stochastic Convergence, Social Learning, Opinion Formation and Decision-Making over Graphs, Emerald Publishing](https://www.emerald.com/books/oa-edited-volume/20922/chapter/107209997/Stochastic-Convergence)
12. [Modes of Convergence, notes by Youyuan Wu](https://youyuanwu.github.io/misc-docs/math/11-probability/06-convergence-modes.html)
13. [STAT 516 Lecture 7: Convergence in Probability and Convergence in Distribution, Purdue University](https://www.stat.purdue.edu/~mlevins/docs/stat516/lecture_notes/Lecture9_STAT517.pdf)
14. [Numerical Methods for Stochastic Differential Equations, Nature Index topic page](https://www.nature.com/nature-index/topics/l4/numerical-methods-for-stochastic-differential-equations)
15. [An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations (Higham), SIAM Review](https://epubs.siam.org/doi/full/10.1137/S0036144500378302)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Convergence of random variables › Relationships among modes of convergence*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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