连续（Continuity）- 有界（Bounded） - 收敛（Convergence）

连续（Continuity）

                 弱                    ---->               强

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continuity     ->    uniform continuity  -> absolute continutiy -> Lipschitz Continuity

连续函数                  一致连续                    绝对连续                   李普希兹连续

   $1/x$                     $\sqrt x$

 

【uniform continutity】

a function f is uniformly continuous if it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.

https://en.wikipedia.org/wiki/Uniform_continuity

 

【absolute continutiy】

| f(x) | is uniformly continuous

https://en.wikipedia.org/wiki/Absolute_continuity

 

【Lipschitz continuity】

函数的变化率是有限的（如果函数可导，则导数有界）

简单来说，Lipschitz连续就类似，一块地不仅没有河流什么的玩意儿阻隔，而且这块地上没有特别陡的坡。其中最陡的地方有多陡呢？这就是所谓的李普希兹常数

https://en.wikipedia.org/wiki/Lipschitz_continuity

 

 

有界（Bounded）

bounded  ->  Uniform boundedness

the sequence of functions $\{ f_n | f_n(x) = sin(nx) \}$ is uniformly bounded

the sequence of functions $\{ g_n | g_n(x) = nsin(x) \}$ is not uniformly bounded

https://en.wikipedia.org/wiki/Uniform_boundedness

 

收敛（Convergence）

逐点收敛(pointwise convergence)  -> 一致收敛(uniform convergence)

【pointwise convergence】

The sequence $f_n(x)$ converges pointwise to the function $f$,  iff

for every $x$, $\lim_{x \to +\infty} f_n=f(x)$

【uniform convergence】

the sequence functions ${ S_n(x) }$ is uniformly convergent:  if for every $\epsilon>0$, there exists a number N, such that for all $n>N$, $|f_n(x)-f(x)|<\epsilon$

https://en.wikipedia.org/wiki/Uniform_convergence

 

随机变量的收敛

研究一列随机变量是否会收敛到某个极限随机变量

https://en.wikipedia.org/wiki/Convergence_of_random_variables

 

【convergence in distribution】

• the weakest form of convergence
• related to central limit theorem

Definition:

A sequence $X_1$, $X_2$, ... of random variables is said to converge in distribution to a random variable X if

$\lim\limits_{n \to \infty} F_n(x)=F(x)$ for every $x\in\mathbb{R}$ at which $F$ is continues. (仅仅考虑$F(x)$连续的地方的分布函数值)

$X_n \overset{d}{\to} X$

 

【Convergence in probability】

• related to the weak law of large numbers
• related to the consistent estimator

meanning：the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.

Definition: A sequence $\{X_n\}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$,  $\lim\limits_{n \to \infty}{Pr(|X_n-X|>\epsilon)}=0$

 $X_n \overset{p}{\to} X$

 

【Almost sure convergence】

 类似于函数列收敛中pointwise convergence，

 Definition：To say that the sequence X_n converges almost surely or almost everywhere or with probability 1 or strongly towards X means that,

$Pr(\lim\limits_{n \to \infty}{X_n=X})=1$

 $X_n \overset{a.s.}{\to} X$