## Convergence in Probability and Convergence with Probability 1

Definations:

For a random vector $X$ and a sequence of random vectors $X_{1},\cdots X_{n},\cdots$ defined on a probability space $(\Omega,\mathcal{F},P)$.

1. Convergence in Probability or Convergence of probability measures, is kind of convergence in measure, in concept of the measure theory.
$X_{n}\stackrel{P}{\to}X$: $P\{||X_{n}-X||>\epsilon\}\to0$ for any fixed $\epsilon>0$.

2. Convergence with Probability 1, is kind of pointwise convergence in real analysis, it is also called almost surely convergence in the measure theory.
$X_{n}\stackrel{a.s.}{\to}X$: if $P\{X_{n}\to X\}=1$ or $P\{X_{n}\nrightarrow X\}=0$.

Convergence in Probability is weaker than Convergence with Probability 1, as stated below:

Theorem: $X_{n}\stackrel{a.s.}{\to}X\Rightarrow X_{n}\stackrel{P}{\to}X$

Proof: $\{X_{n}\to X\}=${$\omega$: for each $\epsilon>0$ there exists an $N(\omega)>0$ such that $||X_{k}(\omega)-X(\omega)||\leq\epsilon$ for all $k>N(\omega)$}

$=${$\cap_{\epsilon>0}${$\omega:$ there exists an $N(\omega)>0$ such that $||X_{k}(\omega)-X(\omega)||\leq\epsilon$ for all $k>N(\omega)$}}

$=${$\cap_{\epsilon>0}\cup_{n=1}^{\infty}${$\omega:$ $||X_{k}(\omega)-X(\omega)||\leq\epsilon$ for all $k>N(\omega)$}}

$=${$\cap_{\epsilon>0}\cup_{n=1}^{\infty}\cap_{k>n}${$\omega:$ $||X_{k}(\omega)-X(\omega)||\leq\epsilon$}}

So $X_{n}\stackrel{a.s.}{\to}X\Longleftrightarrow P\{X_{n}\to X\}=1$

$\Longleftrightarrow P\{\cap_{\epsilon>0}\cup_{n=1}^{\infty}\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1$

$\Longleftrightarrow P\{\cup_{n=1}^{\infty}\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1$ for all $\epsilon>0$

$\Longleftrightarrow\lim_{n\to\infty}P\{\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1$ for all $\epsilon>0$

$\Longleftrightarrow\lim_{n\to\infty}P\{\cup_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||>\epsilon\}\}=0$ for all $\epsilon>0$

$\Rightarrow$$\lim_{n\to\infty}P\{||X_{n}-X||>\epsilon\}=0$ for all $\epsilon>0$, or $P\{||X_{n}-X||>\epsilon\}\to0$ for any fixed $\epsilon>0$, that is $X_{n}\stackrel{P}{\to}X$.

if $X_{n}\stackrel{a.s.}{\to}X$, let the set of unconvergence points as $U_{a.s.}=\{\omega:X_{n}(\omega)\nrightarrow X(\omega)\}$ then $P\{U_{a.s.}\}=0$.
if $X_{n}\stackrel{P}{\to}X$, let the set of unconvergence points as $U_{P}=\{\omega:X_{n}(\omega)\nrightarrow X(\omega)\}$ then we donot have $P\{U_{P}\}=0$.

Convergence in Probability do not require $P\{U_{P}\}=0$ but release the condition to $\lim_{n\to\infty}P\{U_{P}(n)\}=0$, where $U_{P}(n)$ is a set which will go smaller as $n\to\infty$.

From the proof above we can see that $U_{P}=\{\cup_{\epsilon}\cap_{n=1}^{\infty}\cup_{k}\{\omega:||X_{k}(\omega)-X(\omega)||>\epsilon\}\}$ and $U_{P}(n)=\{\omega:||X_{n}-X||>\epsilon\}$, that is to say $U_{P}$ is bigger than $U_{a.s.}$

References:
A Course in Large Sample Theory(Lecture notes), Xianyi Wu
A Course in Large Sample Theory, Thomas S. Ferguson

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