convergence_of_random_variables

# Convergence of Random Variables¶

## Convergence in distribution:¶

Let $F_n(x)$ be the cdf corresponding to r.v. $X_n$ at point $x\in\mathcal{X}$, then convergence in distribution is defined as:

$$\lim_{n\rightarrow\infty}F_n(x)=F(x)$$

## Convergence in probability:¶

For all $\epsilon>0$ we have:

$$\lim_{n\rightarrow\infty}\mathbb{P}(|X_n-X|>\epsilon)=0$$

## Almost sure convergence:¶

$$\mathbb{P}(\lim_{n\rightarrow\infty}X_n=X)=1$$

## Convergence in mean¶

If $\mathbb{E}\left[|X_n|^r\right]$ and $\mathbb{E}\left[|X|^r\right]$ exist, $X_n$ converges to $X$ in the r-th mean if

$$\lim_{n\rightarrow\infty}\mathbb{E}\left[|X_n-X|^r\right]=0$$

Where the most common case is $r=2$ ("convergence in mean-square")