# Convergence in Probability

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

The sequence converges in probability to $Y$ if

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

# Convergence with Probability 1

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

The sequence converges with probability 1 to $Y$ if

$P(\lim _{n\to \infty }X_{n}=Y)=1$ Suppose that a sequence $x_{1},x_{2},...,x_{n}$ is such that

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

$X_{n}{\xrightarrow {P}}Y$ # Convergence in Distribution

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

The sequence converges in distribution to $Y$ if

$\lim _{n\to \infty }P(X_{n}\leq {x})=P(X\leq {x})$ for every real value $x$ such that $P(X=x)=0$ .

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