# Convergence of Random Variable

### Pointwise or sure convergence

A sequence of random variables {X

_{n}}_{n∈N}is said to converge point-wise or surely to X if

X_{n}(ω) → X(ω), ∀ ω∈Ω

### Almost sure convergence

A sequence of random variables {X

_{n}}_{n∈N}is said to converge almost surely or with probability 1 to X if

P ({ω|X_{n}(ω) → X(ω)}) = 1

This convergence is also known as convergence with probability 1. It is denoted by *a.s.* or *w.p.1.* To get the intuition, sure convergence says that the set of ω’s where the random variables converge has a probability one.

### Convergence in probability

A sequence of random variables {X

_{n}}_{n∈N}is said to converge in probability to X if

lim_{n→∞}P(|X_{n}−X|>ε)=0, ∀ ε>0.

This is denoted by *i.p.* or *p*.

### Convergence in r^{th} mean

A sequence of random variables {X

_{n}}_{n∈N}is said to converge in r^{th}mean to X if

lim_{n→∞}E[|Xn − X|^{r}]=0

When r=2 this is called convergence in mean squared/quadratic/L2 mean. This is the most common form of convergence used. This is denoted by *q.m.*

### Convergence in distribution

A sequence of random variables {X

_{n}}_{n∈N}is said to converge in distribution to X if

lim_{n→∞}F_{Xn}(x) = F_{X}(x), ∀ x ∈ R where F_{X}(·) is continuous.

This convergence is also known as weak convergence.