Convergence of Random Variable

• 依分布收敛：$X\_n\stackrel{d}{\to }X$，
• 依概率收敛：$X\_n\stackrel{P}{\to }a$，
• 几乎处处收敛：$X\_n\stackrel{a.s.}{\to }X$，

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Convergence in distribution

written as:

$$X\_n\stackrel{d}{\to }X$$

the distribution converges:

$$\lim \_n\to \infty F\_n(x)\to F(x)$$

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Convergence in Probability

written as:

$$X\_n\stackrel{P}{\to }a$$

$X\_n$ is close to $x\_0$ with greater and greater probability.

$$\lim \_n\to \infty P(|X\_n-x\_0|\ge ϵ)=0$$

given a large $N$, the probability of being outlier will be very small, but still has a chance.

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almost sure convergence

Written as:

$$X\_n\stackrel{a.s.}{\to }X$$

The probability of the random sequence convergence is one.

$$P(\omega \in \Omega :\lim \_n\to \infty X\_n(\omega )=X(\omega ))=1$$

It guarantees that any sample sequence would definitely converges.

收敛于分布$X$的事件概率为一

说的是随机变量序列收敛于某个确定实数的概率为1。