# Lp convergence of random variables
Convergence in \(L_p\) is a mode of convergence of random variables in which the expected p-th power of the error, \(\mathrm{E}[\,|X_n - X|^p\,]\), tends to zero as \(n \to \infty\).
| Key fact | Statement |
|---|---|
| Definition | \(X_n \to X\) in \(L^p\) means \(\lim E[|X_n - X|^p] = 0\), equivalently \(\|X_n - X\|_p \to 0\) [1](https://luc-umass.github.io/html/Part4.html) |
| Precondition | All variables involved must lie in \(L^p\), that is, \(E|X|^p < \infty\) [2](https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf) |
| Implication chain | \(L^p\) convergence implies convergence in probability [1](https://luc-umass.github.io/html/Part4.html) • [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf) |
| Order monotonicity | On a probability space, \(L^{q} \subset L^{p}\) for \(p < q\), so \(L^{q}\) convergence implies \(L^{p}\) convergence [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf) |
| Moment converse | Convergence in probability plus \(\lim E\,|X_{n}|^{p} = E\,|X|^{p}\) gives uniform integrability of \(|X_{n}|^{p}\) and \(L^{p}\) convergence [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf) |
| Counterexample | \(f_n = n\) on (0, 1/n) converges to 0 almost surely but \(E|f_n|^p = n^{p-1}\) for every n [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf) |
| Completeness | \(L^{p}\) is a complete separable metric space for every \(p \geq 1\) [2](https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf) |
## Definition and basic properties
Write \(L^p(\Omega, F, P)\) for the vector space of random variables \(X\) on a probability space with \(E|X|^p < \infty\). The \(p = 1\) members are the integrable variables with finite means, and the \(p = 2\) members are the square-integrable variables with finite variance [2](https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf). A sequence \((X_n)\) converges in \(L^p\) to \(X \in L^p(\Omega)\) if \(\lim E[|X_n - X|^p] = 0\) [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf); since \(E|Y|^p = \|Y\|_p^p\), this is the same as \(\|X_n - X\|_p \to 0\) [1](https://luc-umass.github.io/html/Part4.html).
For \(p \geq 1\), Minkowski's inequality makes \(\|X\|_{p}\) a norm, and the induced metric \(d(X, Y) = \|X - Y\|_{p}\) turns \(L^{p}\) into a complete separable metric space [2](https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf).
[Pointwise convergence](https://www.edgepedia.org/pointwise-convergence) of a sequence of \(L^{p}\) functions does not by itself put the limit in any \(L^{p}\): the functions \(f_{n}(\omega) = 0\) for \(\omega < 1/n\) and \(1/\omega\) for \(\omega \geq 1/n\) converge almost surely to \(1/\omega\), which belongs to no \(L^{p}\) on the unit interval [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf).
## Criteria via moment inequalities
Several standard inequalities convert moment bounds into convergence statements. [Hölder's inequality](https://www.edgepedia.org/holders-inequality) for conjugate indices \(p, q \in (1, \infty)\) and Minkowski's inequality for \(p \in [1, \infty)\) are the core tools underlying the \(L^p\) norm structure [6](https://www.statslab.cam.ac.uk/~james/Lectures/pm13.pdf).
**Domination** gives the most-used sufficient condition: if \(X_n \to X\) in probability and \(|X_n| \le Y\) for some \(Y \in L^p\), then \(X_n \to X\) in \(L^p\) [1](https://luc-umass.github.io/html/Part4.html). This is the random-variable form of the dominated convergence criterion, and it verifies \(L^p\) convergence from a pointwise or in-probability limit plus a fixed p-th-moment ceiling. At the other end, [Markov's inequality](https://www.edgepedia.org/markovs-inequality) \(P[|X_n - X| \ge \varepsilon] \le E[|X_n - X|^p]/\varepsilon^p\) is what lets an established \(L^p\) limit be downgraded to convergence in probability [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf).
## How it compares with other modes of convergence
The implications run one way. \(L^{p}\) convergence implies convergence in probability by Markov's inequality [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf) • [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf).
Neither converse holds without extra hypotheses.
**\(L^p\) does not imply almost sure convergence.** On [0, 1], the sliding-block indicator sequence 1, \(I_{(0,1/2]}\), \(I_{(1/2,1]}\), \(I_{(0,1/3]}\), \(I_{(1/3,2/3]}\), … converges to 0 in \(L^p\) for all finite \(p\), yet every point \(\omega\) has \(f_n(\omega) = 1\) infinitely often, so there is no almost sure convergence [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf).
**Almost sure convergence does not imply L^p convergence.** Take \(f_n = n\) on \((0, 1/n)\) and 0 elsewhere: \(f_n \to 0\) almost surely and in probability, but \(E|f_n|^p = n^{p-1}\) for all finite p [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf). The USC notes systematize such separations with the growing pulse family \(\xi_n(x) = n^{\alpha} I(0 < x < n^{-\beta})\), whose exponents α and β can be tuned to converge in some modes and not others [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf).
**Convergence in probability does not imply \(L^p\) convergence.** A modified sliding-hump sequence (\(Y_1 = X_1\), \(Y_2 = 2 \cdot X_2\), \(Y_3 = 2 \cdot X_3\), …) still converges in probability to 0 but has \(E[Y_n] = 1\), so it cannot converge to 0 in \(L^1\) [1](https://luc-umass.github.io/html/Part4.html). What restores the converse is domination: \(|X_n| \leq Y \in L^p\) upgrades convergence in probability to \(L^p\) convergence [1](https://luc-umass.github.io/html/Part4.html).
## Comparing orders p
On a probability space, if \(p < q\) then Hölder's inequality gives \(\|X\|_{p} \leq \|X\|_{q}\), so \(L^{q}(\Omega) \subset L^{p}(\Omega)\), and convergence in \(L^{q}\) implies convergence in \(L^{p}\) [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf). Equivalently, since the \(L^{p}\) norms are increasing in \(p\), convergence in \(L^{p}\) implies convergence in \(L^{r}\) for every \(r < p\) [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf). Stronger convergence at a higher moment order therefore cascades downward for free.
The reverse direction needs a bound on the variables. The two most used orders have distinct geometries: \(L^{1}\) controls expectations, while \(L^{2}\) is a self-dual [Hilbert space](https://www.edgepedia.org/hilbert-space), a point taken up under applications.
## Moments and the boundary with uniform integrability
\(L^1\) convergence preserves expectations, because \(|E[X_n] - E[X]| \leq E[|X_n - X|]\) [1](https://luc-umass.github.io/html/Part4.html).
The converse is the interesting half. If ξ_n → ξ in probability and lim E|ξ_n|^p = E|ξ|^p for some p > 0, then the family {|ξ_n|^p} is uniformly integrable and ξ_n → ξ in L^p [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf). [Uniform integrability](https://www.edgepedia.org/uniform-integrability) here means lim as a → ∞ of sup over n of E(|ξ_n| I(|ξ_n| > a)) = 0 [3](https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf). In L^p terms, a class F ⊂ L^p(Ω) is uniformly integrable in L^p when for every ε > 0 there exists K bounding the tail contributions uniformly [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf).
Where this article stops: pushing the converse beyond the moment condition above leads to the notion of uniform integrability [4](https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf), and the full Vitali theory of uniform integrability, its criteria, and its equivalence theorems, is treated in its own article. What remains here is the delimitation: the growing-pulse and sliding-hump examples above show concretely that without a uniform tail bound the implication from convergence in probability to \(L^p\) fails.
## By the numbers
The counterexample family is fully computable. For \(f_n = n\) on \((0, 1/n)\): \(E|f_n|^p = n^p \cdot (1/n) = n^{p-1}\), which equals 1 for \(p = 1\) and diverges for \(p > 1\), so in either case it does not tend to zero and the a.s. limit 0 is not approached in any \(L^p\) [5](https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf). For the modified sliding-hump sequence, \(E[Y_n] = 1\) for every n even though \(Y_n \to 0\) in probability, separating first moments from in-probability limits [1](https://luc-umass.github.io/html/Part4.html).
**Weak-law rates** are the classical home of \(L^p\) convergence criteria. Markov's theorem asserts, under suitable conditions, the \(L^2\)-vanishing of \(n^{-2}\)-scaled partial sums, while Khintchine's theorem gives \(L^1\)-vanishing of sample means of centered \(L^1\) variables under conditions controlling dependence structure and distributional homogeneity [7](https://doi.org/10.46298/cm.10292). A recent generalization removes dependence assumptions entirely: if \((1/a_n) \sum_{i=1}^{b_n} |X_i|_{L^p} \to 0\) for unbounded integer sequences \((a_n)\), \((b_n)\), then \(a_n^{-1} \sum (X_i - E X_i) \to 0\) in \(L^p\) for \(1 \le p \le \infty\) [7](https://doi.org/10.46298/cm.10292). As an explicit moment computation in the same spirit, the unbiased sample variance of iid variables satisfies \(E[V_n] = \sigma^2\) [1](https://luc-umass.github.io/html/Part4.html).
## Applications and practice
\(L^{2}\) is a self-dual real Hilbert space with inner product \(E[X \cdot Y]\) [2](https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf).
**Martingale convergence theorems** are a principal application of \(L^p\) convergence: a martingale \((X_n)\) converges in \(L^p\) to a limit \(X_{\infty}\) when \(\|X_n - X_{\infty}\|_p \to 0\), and by [Chebyshev's inequality](https://www.edgepedia.org/chebyshevs-inequality) such convergence implies convergence in probability [8](https://people.math.wisc.edu/~roch/grad-prob/gradprob-notes16.pdf). In limit theorems, the \(L^p\) rates above enter weak laws and, via estimation of observational time series whose means vanish on average due to exogenous structural factors, structural equation modeling [7](https://doi.org/10.46298/cm.10292).
## References
1. Part 4: Convergence of random variables and limit theorems (UMass) — https://luc-umass.github.io/html/Part4.html
2. Lecture notes on Expectation Inequalities and Lp Spaces, Duke STA 711 — https://www2.stat.duke.edu/courses/Fall20/sta711/lec/wk-05.pdf
3. Modes of convergence of random variables (USC) — https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2025/11/RV-Convergence.pdf
4. Convergence of random variables and large deviations (Université Paris-Saclay) — https://www.imo.universite-paris-saclay.fr/~pierre-loic.meliot/convergence/convergence.pdf
5. Lecture notes 5: Lp spaces and modes of convergence, CMU 36-710 — https://www.stat.cmu.edu/~arinaldo/Teaching/36710-36752/Lecture_Notes/lec_notes_5.pdf
6. Probability and Measure lecture notes: Hölder and Minkowski inequalities (Cambridge) — https://www.statslab.cam.ac.uk/~james/Lectures/pm13.pdf
7. A General Weak Law of Large Numbers for Sequences of Lp Random Variables — https://doi.org/10.46298/cm.10292
8. Notes 16: Martingales in Lp (UW–Madison) — https://people.math.wisc.edu/~roch/grad-prob/gradprob-notes16.pdf
9. Concentration of norms of random vectors with independent p-sub-exponential coordinates, Archiv der Mathematik (2025) — https://link.springer.com/article/10.1007/s00013-025-02199-7
10. Mean convergence for Banach space-valued random elements indexed in measure spaces (arXiv, 2026) — https://arxiv.org/abs/2608.30357
11. Martingale central limit theorems in p-Wasserstein distance (arXiv, 2026) — https://arxiv.org/abs/2609.11557
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Convergence of random variables › Lp convergence of random variables*
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