Four modes of convergence
Section of Expectation and convergence, part of Stochastic processes
Four modes of convergence
| Type | Section |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
Definition 1 (Modes of convergence)
- X_n\to X almost surely when \mathrm{P}\{\omega: X_n(\omega)\to X(\omega)\}=1.
- X_n\to X in probability when \mathrm{P}(|X_n-X|>\varepsilon)\to 0 for every \varepsilon>0.
- X_n\to X in L^p, for p\ge 1, when \mathbb{E}|X_n-X|^p\to 0.
- X_n\to X in distribution when F_{X_n}(x)\to F_X(x) at every continuity point x of F_X.
The arrows that hold in general are
L^p \;\Rightarrow\; \text{in probability} \;\Rightarrow\; \text{in distribution}, \qquad \text{almost surely} \;\Rightarrow\; \text{in probability}.
L^p convergence implies convergence in probability by Markov’s inequality applied to |X_n-X|^p:
\mathrm{P}(|X_n-X|>\varepsilon)\le\frac{\mathbb{E}|X_n-X|^p}{\varepsilon^p}.
Almost-sure convergence implies convergence in probability because the event \{|X_n-X|>\varepsilon\text{ infinitely often}\} is a tail event of probability 0, so the probability of the nth deviation tends to 0. Convergence in probability implies convergence in distribution. The reverse arrows fail in general, and the examples below locate the missing assumption.
One reverse arrow does hold. If X_n converges in distribution to a constant c, then X_n converges to c in probability. A distributional limit that is not a constant is a statement about laws, not about closeness of X_n(\omega) and X(\omega) on a shared sample space. A copy X_n with exactly the same distribution as X, built independently of X, converges in distribution to X and does not converge to X in probability.
Convergence in distribution moves probability on the line. It does not say that X_n-X is small. Convergence of expectations needs a separate hypothesis. Dominated convergence and uniform integrability are two such hypotheses.
Used in
See also
Monotone convergence, Fatou, and dominated convergence · Modes of convergence · Strong law · Uniform integrability · Inequalities and limits under the expectation · Where the reverse implications fail · Laws of large numbers · Characteristic functions