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")