01 - Convergence of RVs

Almost Surely

Let $X_1,X_2,\cdots$ and $X$ be random variables

$X_n\to X$ almost surely (or with probability $1$) if

$$\mathbb{P}(X_n\to X\text{ as }n\to \infty )=1$$

Then we can abbreviate it as

$$X_n\to X\text{ a.s.}$$

Event Statement

$$\{X_n\to X\text{ as }n\to \infty \}\text{ is an event}$$

More formally it is

$$\{\omega \in \Omega :X_n(\omega )\to X(\omega )\text{ as }n\to \infty \}$$

In Probability

Let $X_1,X_2,\cdots$ and $X$ be random variables

$X_n\to X$ in probability if

$$\forall ϵ>0:\mathbb{P}(|X_n-X|<ϵ)\to 1\text{ as }n\to \infty$$

Then we can abbreviate it as

$$X_n\stackrel{P}{\to }X\text{ as }n\to \infty$$

In Distribution (Weakly)

Let $F_1,F_2,\cdots$ and $F$ be the distribution of $X_1,X_2,\cdots$ and $X$ respectively

$X_n\to X$ in distribution if
For every $x$ where $F$ is continuous at $x$ then

$$F_n(x)\to F(x)\text{ as }n\to \infty$$

Then we can abbreviate it as

$$X_n\stackrel{d}{\to }X\text{ as }n\to \infty$$