Collection of standard facts about convergence of random variables

Question

The goal of this question is to collect standard general facts about convergence of random variables (in $\mathbb L^p$, in probability, in distribution) in order to use them when answering questions. I am aware of the existence of this meta-thread but here the goal is more to have directly available proofs than to list repeated questions. Usually, when typing an answer, we use some "well-known" results but it is hard to find proofs on the site.

The answer should mention the fact (for example, convergence in probability implies convergence in distribution) and either its proof or a link to a thread giving a complete proof. I know that the list could be very big so we can try to restrict to the most general fact.

Answer 1

• $L^p$-convergence implies convergence in probability.

• Partial converse: If $X_n \to X$ in probability and $(X_n)_{n \in \mathbb{N}}$ is $L^p$-bounded, then $X_n \to X$ in $L^q$ for $q<p$.

• If $X_n \to X$ almost surely and $\mathbb{E}(|X_n|)\to \mathbb{E}(|X|)<\infty$, then $X_n \to X$ in $L^1$.

• Counterexamples for implications which fail in general to hold (a.e. convergence does not imply convergence in probability and so on)

• If $X_n \stackrel{d}{\to} X$ and $Y_n \stackrel{\mathbb{P}}{\to} c$ for a constant $c$, then $X_n Y_n \stackrel{d}{\to} cX$. (Slutsky's theorem I). Note: If $c=0$ then we even get convergence in probability (since $X_n Y_n \stackrel{d}{\to} 0=cX$ implies $X_n Y_n \stackrel{\mathbb{P}}{\to} 0$), see also this question.

• If $X_n \stackrel{d}{\to} X$ and $Y_n \stackrel{d}{\to} c$ for a constant $c$, then $X_n+Y_n \stackrel{d}{\to} X+c$. (Slutsky's theorem II)

• $X_n \stackrel{d}{\to} X$ and $Y_n \stackrel{d}{\to} Y$ does, in general, not imply $X_n+Y_n \stackrel{d}{\to} X+Y$.

• If $X_n \to X$ and $Y_n \to Y$ in probability, then the product $X_n Y_n$ converges in probability to $X \cdot Y$.

• If $X_n \to c \neq 0$ in probability, then $1/X_n \to 1/c$ in probability.

• If $X_n \to X$ in distribution and $c_n \to c$, then $c_n X_n \to cX$ in distribution.

• If $X_n \to X$ in distribution and $X_n$ is Gaussian for each $n \in \mathbb{N}$, then $X$ is Gaussian.

• If $X_n$ is a Cauchy sequence in measure, then there exists a random variable $X$ such that $X_n \to X$ in probability.

• $X_n \to X$ in probability if, and only if, $\mathbb{E}\min\{|X_n-X|,1\} \to 0$ as $n \to \infty$.

• Convergence in probability implies almost everywhere convergence of a subsequence.

• If $S_n = \sum_{j=1}^n X_j$ is a sum of independent random variables which converges in probability, then $S_n$ converges almost surely (Lévy's equivalence theoem).

• If $S_n = \sum_{j=1}^n X_j$ is a sum of independent random variables which converges in distribution, then $S_n$ converges almost surely.

• Almost sure convergence is not metrizable.

Answer 2

1. Convergence in probability implies convergence in distribution.
2. The converse is true if the limit is a constant.
3. Convergence in probability is preserved by continuous maps.
4. Here is a thread dealing with a link between almost sure convergence and convergence in $\mathbb L^1$. It shows that if $\lim_{n \rightarrow \infty} \mathbb{E}|X_n| \rightarrow \mathbb{E}|X| < \infty$, then $X_n \rightarrow X$ in $L^1$ i.e. $\Bbb E[|X_n-X|] \to 0$.
5. If $(X_n)$ and $X$ are random variables such that $X_n \to X$ in distribution and such that $\{X_n\mid n\geq 1\}$ is uniformly integrable, then $E[X_n]\to E[X]$.
6. Combining facts 1., 4. and 5. we get after a reasoning on subsequences that convergence in probability combined with uniform integrability is equivalent to convergence in $\mathbb L^1$.

Answer 3

1. Convergence a.s. implies almost uniform convergence by Egoroff's Theorem. Similarly, convergence in probability will imply almost uniform convergence of a subsequence.