Subjects: Probability Theory
Links: Random Variables, Lp spaces, Convergence of Measurable Functions

Let $X_1,\dots ,X_n,\dots$ be a sequence of random variables. There are a lot of types of convergence

Types of Convergence

Pointwise Convergence

It is the absolute simplest.

Let $X_n$ be the sequence of random variables, we say that it converges pointwise if for all $\omega \in \mathrm{\Omega }$, $$\lim_{n \to \infty} X_n(\omega) = X(\omega)$$
By some results in measure theory, i think that $X$ will always be a random variable.

We can denote it as $X_n\to X$, or if we want to specify, $X_n\stackrel{p}{⟶}X$

Almost Everywhere Convergence

The sequence of random variables $X_n$, converges converges almost surely to $X$ if $$P({\omega \in \Omega \mid \lim_{n \to \infty} X_n(\omega) = X(\omega)} = 1$$or $$P(\lim_{n\to \infty} X_n = X) = 1$$We can denote is as $X_n\stackrel{a.s.}{⟶}X$, or $\underset{n\to \mathrm{\infty }}{lim}{X}_n=Xa.s.$

Convergence in Probability

The sequence of random variables $X_n$ converges to $X$ in probability if for every $\epsilon >0$, $$P({\omega\in \Omega \mid |X_n(\omega) - X(\omega)|>\varepsilon})=0 $$
We can denote this kind of convergence by $X_n\stackrel{P}{⟶}X$, omitting $\omega$. The condition is $$P(\lim_{n \to \infty} |X_n-X| > \varepsilon) = 0 $$

Convergence in Mean

The sequence of random variables $X_n$ converges to $X$ in mean if $$\lim_{n \to \infty} E|X_n - X| = 0$$
This type of convergence is also called $L^1$ convergence and it's denoted as $X_n\stackrel{L^1}{⟶}X$

Convergence in Mean Squared

The sequence of random variables $X_n$ converges to $X$ in mean squared if $$\lim_{n \to \infty} E|X_n - X|^2 = 0$$
This type of convergence is also called $L^1$ convergence and it's denoted as $X_n\stackrel{L^2}{⟶}X$

Convergence in $L^p$

The sequence of random variables $X_n$ converges to $X$ in $L^p$ if $$\lim_{n \to \infty} E|X_n - X|^p = 0$$
This type of convergence is also called $L^1$ convergence and it's denoted as $X_n\stackrel{L^p}{⟶}X$

Convergence in distribution

The sequence of random variables $X_n$ converges to $X$ in distribution if for all $x$ where the function $F_X$ is continuous. it satisfies that $$\lim_{n \to \infty} F_{X_n}(x) = F_X(x)$$
Where $F_{X_n}$ is the cdf of $X_n$, and $F_X$ being the cdf $X$. We denote denote it as $X_n\stackrel{d}{⟶}X$, $X_n\stackrel{\mathcal{D}}{⟶}X$, or lastly $F_{X_n}\stackrel{d}{⟶}{F}_X$. This type of convergence is also known as weak convergence since it less restrictive than the others.

Relations between types of convergence

• Almost sure convergence $⟹$ Convergence in probability
• Convergence in probability $⟹⧸\phantom{⟹}$ Almost sure convergence (in general)
• Almost sure convergence $⟹⧸\phantom{⟹}$ Almost sure convergence in $L^1$ (in general)
• Convergence in $L^2$ $⟹$ Convergence in $L^1$
• Convergence in $L^1$ $⟹$ Convergence in probability
• Convergence in probability $⟹$ Convergence in distribution

Important Theorems

Monotone Convergence Theorem

Let $0\le X_1\le X_2\le \cdots$ be an increasing sequence of random variables that converges almost surely to $X$. Then $$\lim_{n \to \infty} E[X_n] = E[X] $$

Dominated Convergence Theorem

Let $X_1,X_2,\dots$ be a sequence of random variables, and a random variable $Y$, that $|X_n|\le Y$ , for all $n\in {\mathbb{N}}^{+}$. If $\underset{n\to \mathrm{\infty }}{lim}{X}_n=Xa.s.$ then $X$ and $X_n$ are integrable and $$\lim_{n \to \infty} E[X_n] = E[X] $$