# Convergence in Probability

Consider a sequence of random variables ${\displaystyle x_{1},x_{2},...,x_{n}}$ and another random variable ${\displaystyle Y}$.

The sequence converges in probability to ${\displaystyle Y}$ if

${\displaystyle \lim _{n\to \infty }P(|X_{n}-Y|\geq \epsilon )=0}$

for every ${\displaystyle \epsilon >0}$.

# Convergence with Probability 1

Consider a sequence of random variables ${\displaystyle x_{1},x_{2},...,x_{n}}$ and another random variable ${\displaystyle Y}$.

The sequence converges with probability 1 to ${\displaystyle Y}$ if

${\displaystyle P(\lim _{n\to \infty }X_{n}=Y)=1}$

Suppose that a sequence ${\displaystyle x_{1},x_{2},...,x_{n}}$ is such that

${\displaystyle X_{n}{\xrightarrow {a.s.}}Y}$

then

${\displaystyle X_{n}{\xrightarrow {P}}Y}$

# Convergence in Distribution

Consider a sequence of random variables ${\displaystyle x_{1},x_{2},...,x_{n}}$ and another random variable ${\displaystyle Y}$.

The sequence converges in distribution to ${\displaystyle Y}$ if

${\displaystyle \lim _{n\to \infty }P(X_{n}\leq {x})=P(X\leq {x})}$

for every real value ${\displaystyle x}$ such that ${\displaystyle P(X=x)=0}$.

If ${\displaystyle X_{n}{\xrightarrow {a.s.}}X}$ then ${\displaystyle X_{n}{\xrightarrow {D}}Y}$

If ${\displaystyle X_{n}{\xrightarrow {P}}Y}$ then ${\displaystyle X_{n}{\xrightarrow {D}}Y}$